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A collection of 19 posts
iPhone AI: Artificial Intelligence Features in Apple iPhone
Apple is dominating the technology sector with its iPhone AI features and apps, making its users' lives more secure and easier than ever. The release of iPhone models over time confirms that Apple uses Artificial Intelligence in iPhones and visions AI as the key factor for its tech supremacy.
ThinkML Team 2 Oct 2022 • 11 min read
10 Best Robot Dogs: Pros and Cons
Robot dogs are taking the world by storm, but is having one a good idea? If you're so excited to buy them but confused about choosing, you'll be thrilled by this blog. We've provided information on how much they cost and what makes them special. Let's take a closer look at our 10 best robot dogs.
ThinkML Team 29 Sep 2022 • 22 min read
AI Dungeon Game - Top 15 AI Dungeon Alternatives
A ground-breaking online text adventure game, AI Dungeon, by Latitude, showcased a novel human-machine collaboration. The game is regarded as the greatest text-based gameplay that offers an unpredictable playing experience to users once they type the dialogue they desire their characters to say.
What Do Artificial Intelligence Experts Predict for 2023?
Let's take an in-depth look at some Artificial Intelligence expert predictions for 2023 and what they could mean for our daily lives. As a society, we have already seen some surprising advancements this year, so now it's time to discover where we're going with "smart thinking".
ThinkML Team 26 Aug 2022 • 6 min read
Is CryptoCompare the Right Way to Invest? Features and Specifications
CryptoCompare is an international cryptocurrency market and one of the leading global cryptocurrency sites, but is it any good? It provides corporate and trade business people with actual-time, outstanding performance, and general data about this particular market and its trading methods.
ThinkML Team 7 Aug 2022 • 10 min read
Essential Tips To Embrace AI Technology For Enhancing Your Business
AI can automate processes and use data to help a business make better, smarter, and faster decisions. It automates repetitive tasks and lessens your reliance on humans. But you cannot introduce technology without knowing it, so here are some crucial pointers for using AI in your organization.
ThinkML Contributor
ThinkML Contributor 30 Jun 2022 • 4 min read
Top Emerging Computer Vision Trends 2022
One of the most powerful and effective types of AI is computer vision. It aims to replicate the complicated human vision system in order for computers to detect and identify items in movies and photos in the same way that humans do.
ThinkML Team 7 Apr 2022 • 11 min read
Top AI Achievements of 2021
AI has proven to be a game-changer in the last decades. It provides extraordinary advantages and has potential for future success in automation. Some of the top AI achievements of 2021 include facial identification, text generation, speech recognition, drug discovery, and automated translation.
Artificial Intelligence in HR
Various corporations and businesses are rapidly adopting AI in order to boost their profits. Artificial intelligence in HR has the potential to solve problems in practically every business, including sales and management.
Can AI Predict Heart Diseases?
Cardiovascular diseases are the leading death cause worldwide. Many artificial medical intelligence researchers are working on improving disease diagnosis, detection, and treatment of CVD. In medicine, AI aims gives unique ways, such as signalling, visual patterns, and waveforms for better analysis.
Top 10 Leading Universities in AI Research
Deep learning, natural language processing, data analytics, and big-data mining are fields of Artificial Intelligence (AI), and many companies are looking for professionals in these fields. A professional degree in AI from a reputed university will help you get started in this industry.
Data Science Interview Questions
Preparing for a data scientist interview is tough since the questions you will be asked about data science are unclear. An interviewer may surprise you with a set of unexpected questions, regardless of how much job experience you have or what data science credentials you possess.
ThinkML Team 6 Aug 2021 • 9 min read
This is the book to read if you want to ace your data scientist interview. This pdf book contains data science interview questions and answers that are gathered from Facebook, Amazon, Apple, Netflix, Microsoft, and Google.
ThinkML Team 27 Jul 2021 • 2 min read
Boost Debt Collection and Recovery using Machine Learning [part 4/5]
This is part-4 of the case study on Boost Debt Collections and Recoveries using Machine Learning (MLBR). A machine learning predictive model to enhance the current recovery system by creating focus groups for business to boost debt collection. In this part, we did comparison of various models
M Umer Mirza 30 Sep 2020 • 3 min read


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"The house which I am about to build will be great, for greater is our God than all the gods. "But who is able to build a house for Him, for the heavens and the highest heavens cannot contain Him? So who am I, that I should build a house for Him, except to burn incense before Him? "Now send me a skilled man to work in gold, silver, brass and iron, and in purple, crimson and violet fabrics, and who knows how to make engravings, to work with the skilled men whom I have in Judah and Jerusalem, whom David my father provided. Share Your Faith Products Canvas Art
SWA sells only 100% ready to go wall decals made out of the highest quality materials. And while it can take up to a month to get a cheap vinyl wall decal from China that doesn't even work, we typically get your wall art to you in less than a week. Customers near us, often get them the next day. That's because we ship most orders the same business day when ordered before noon PST. Christian Art and Gifts
"See, I have called by name Bezalel the son of Uri, son of Hur, of the tribe of Judah, and I have filled him with the Spirit of God, with ability and intelligence, with knowledge and all craftsmanship, to devise artistic designs, to work in gold, silver, and bronze, in cutting stones for setting, and in carving wood, to work in every craft. And behold, I have appointed with him Oholiab, the son of Ahisamach, of the tribe of Dan. And I have given to all able men ability, that they may make all that I have commanded you: ... Christian Canvas Art
© Kim Jones and Salvaged Living, 2019. Unauthorized use and/or duplication of this material without express and written permission from this blog's author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Hunt & Host with appropriate and specific direction to the original content. Please only use one picture with a link back unless individual permission is given otherwise by Salvaged Living. Full disclosure here. Bible Scripture Verse Art
He also made two capitals of molten bronze to set on the tops of the pillars; the height of the one capital was five cubits and the height of the other capital was five cubits. There were nets of network and twisted threads of chainwork for the capitals which were on the top of the pillars; seven for the one capital and seven for the other capital. So he made the pillars, and two rows around on the one network to cover the capitals which were on the top of the pomegranates; and so he did for the other capital. The capitals which were on the top of the pillars in the porch were of lily design, four cubits. There were capitals on the two pillars, even above and close to the rounded projection which was beside the network; and the pomegranates numbered two hundred in rows around both capitals. Thus he set up the pillars at the porch of the nave; and he set up the right pillar and named it Jachin, and he set up the left pillar and named it Boaz. On the top of the pillars was lily design. So the work of the pillars was finished. Now he made the sea of cast metal ten cubits from brim to brim, circular in form, and its height was five cubits, and thirty cubits in circumference. Under its brim gourds went around encircling it ten to a cubit, completely surrounding the sea; the gourds were in two rows, cast with the rest. It stood on twelve oxen, three facing north, three facing west, three facing south, and three facing east; and the sea was set on top of them, and all their rear parts turned inward. It was a handbreadth thick, and its brim was made like the brim of a cup, as a lily blossom; it could hold two thousand baths. Then he made the ten stands of bronze; the length of each stand was four cubits and its width four cubits and its height three cubits. This was the design of the stands: they had borders, even borders between the frames, and on the borders which were between the frames were lions, oxen and cherubim; and on the frames there was a pedestal above, and beneath the lions and oxen were wreaths of hanging work. Now each stand had four bronze wheels with bronze axles, and its four feet had supports; beneath the basin were cast supports with wreaths at each side. Its opening inside the crown at the top was a cubit, and its opening was round like the design of a pedestal, a cubit and a half; and also on its opening there were engravings, and their borders were square, not round. The four wheels were underneath the borders, and the axles of the wheels were on the stand. And the height of a wheel was a cubit and a half. The workmanship of the wheels was like the workmanship of a chariot wheel. Their axles, their rims, their spokes, and their hubs were all cast. Now there were four supports at the four corners of each stand; its supports were part of the stand itself. On the top of the stand there was a circular form half a cubit high, and on the top of the stand its stays and its borders were part of it. He engraved on the plates of its stays and on its borders, cherubim, lions and palm trees, according to the clear space on each, with wreaths all around. He made the ten stands like this: all of them had one casting, one measure and one form. He made ten basins of bronze, one basin held forty baths; each basin was four cubits, and on each of the ten stands was one basin. Then he set the stands, five on the right side of the house and five on the left side of the house; and he set the sea of cast metal on the right side of the house eastward toward the south. Now Hiram made the basins and the shovels and the bowls. So Hiram finished doing all the work which he performed for King Solomon in the house of the LORD: the two pillars and the two bowls of the capitals which were on the top of the two pillars, and the two networks to cover the two bowls of the capitals which were on the top of the pillars; and the four hundred pomegranates for the two networks, two rows of pomegranates for each network to cover the two bowls of the capitals which were on the tops of the pillars; and the ten stands with the ten basins on the stands; and the one sea and the twelve oxen under the sea; and the pails and the shovels and the bowls; even all these utensils which Hiram made for King Solomon in the house of the LORD were of polished bronze. In the plain of the Jordan the king cast them, in the clay ground between Succoth and Zarethan. Solomon left all the utensils unweighed, because they were too many; the weight of the bronze could not be ascertained. Solomon made all the furniture which was in the house of the LORD: the golden altar and the golden table on which was the bread of the Presence; and the lampstands, five on the right side and five on the left, in front of the inner sanctuary, of pure gold; and the flowers and the lamps and the tongs, of gold; and the cups and the snuffers and the bowls and the spoons and the firepans, of pure gold; and the hinges both for the doors of the inner house, the most holy place, and for the doors of the house, that is, of the nave, of gold. Thus all the work that King Solomon performed in the house of the LORD was finished And Solomon brought in the things dedicated by his father David, the silver and the gold and the utensils, and he put them in the treasuries of the house of the LORD. Bible Scripture Verse Art
"You shall make two cherubim of gold, make them of hammered work at the two ends of the mercy seat. "Make one cherub at one end and one cherub at the other end; you shall make the cherubim of one piece with the mercy seat at its two ends. "The cherubim shall have their wings spread upward, covering the mercy seat with their wings and facing one another; the faces of the cherubim are to be turned toward the mercy seat.read more. Christian Canvas Art
He made an altar of bronze, twenty cubits long and twenty cubits wide and ten cubits high. Then he made the sea of cast metal. It was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured its circumference. Under it were figures of gourds, for ten cubits, compassing the sea all around. The gourds were in two rows, cast with it when it was cast. It stood on twelve oxen, three facing north, three facing west, three facing south, and three facing east. The sea was set on them, and all their rear parts were inward. Its thickness was a handbreadth. And its brim was made like the brim of a cup, like the flower of a lily. It held 3,000 baths. ...
Now when the wall had been built and I had set up the doors, and the gatekeepers, the singers, and the Levites had been appointed, I gave my brother Hanani and Hananiah the governor of the castle charge over Jerusalem, for he was a more faithful and God-fearing man than many. And I said to them, "Let not the gates of Jerusalem be opened until the sun is hot. And while they are still standing guard, let them shut and bar the doors. Appoint guards from among the inhabitants of Jerusalem, some at their guard posts and some in front of their own homes." The city was wide and large, but the people within it were few, and no houses had been rebuilt. Then my God put it into my heart to assemble the nobles and the officials and the people to be enrolled by genealogy. And I found the book of the genealogy of those who came up at the first, and I found written in it: ... Christian Canvas Art
David built houses for himself in the city of David. And he prepared a place for the ark of God and pitched a tent for it. Then David said that no one but the Levites may carry the ark of God, for the Lord had chosen them to carry the ark of the Lord and to minister to him forever. And David assembled all Israel at Jerusalem to bring up the ark of the Lord to its place, which he had prepared for it. And David gathered together the sons of Aaron and the Levites: of the sons of Kohath, Uriel the chief, with 120 of his brothers; ...
Comfort, comfort my people, says your God. Speak tenderly to Jerusalem, and cry to her that her warfare is ended, that her iniquity is pardoned, that she has received from the Lord's hand double for all her sins. A voice cries: "In the wilderness prepare the way of the Lord; make straight in the desert a highway for our God. Every valley shall be lifted up, and every mountain and hill be made low; the uneven ground shall become level, and the rough places a plain. And the glory of the Lord shall be revealed, and all flesh shall see it together, for the mouth of the Lord has spoken." ... Christian Art and Gifts
Our Gift Shop is full of amazing products that we think you might like that compliment many of our wall arts. All the items in our Gift Shop are made by other companies. When you click on one to purchase you will be directed to Amazon to make the purchase. Please note that those purchases are not covered by our guarantees. All purchases, refunds, exchanges, or service issues must be dealt with through Amazon. However, at no cost to you, we make a small commission when you purchase through our links. We greatly appreciate that support as it allows us to help keep the price on our wall arts down. Thank you in advance for any purchases you make through our Gift Shop. Christian Canvas Art
This verse is a reminder for me that all I need is Christ. When I feel helpless or weak, I don't primarily need a practical solution or answer. I need God's grace. God's grace is sufficient. It is all I need. His strength shines brightly when I am weak and depending on Him. I don't need my life to be in perfect order or all to be going well in the world around me. All I need is Christ. Christian Art and Gifts
He also made two pillars for the front of the house, thirty-five cubits high, and the capital on the top of each was five cubits. He made chains in the inner sanctuary and placed them on the tops of the pillars; and he made one hundred pomegranates and placed them on the chains. He erected the pillars in front of the temple, one on the right and the other on the left, and named the one on the right Jachin and the one on the left Boaz. Share Your Faith Products Canvas Art
So he made two doors of olive wood, and he carved on them carvings of cherubim, palm trees, and open flowers, and overlaid them with gold; and he spread the gold on the cherubim and on the palm trees. So also he made for the entrance of the nave four-sided doorposts of olive wood and two doors of cypress wood; the two leaves of the one door turned on pivots, and the two leaves of the other door turned on pivots. He carved on it cherubim, palm trees, and open flowers; and he overlaid them with gold evenly applied on the engraved work. Christian Canvas Art

redpajama_set_name: RedPajamaCommonCrawl


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Congo Ebola center set on fire after armed attack
By Latest Media News on February 27, 2019
GOMA, Democratic Republic of Congo (Reuters) – Armed assailants attacked an Ebola treatment center in eastern Democratic Republic of Congo on Wednesday, setting off a fire and becoming embroiled in an extended gunbattle with security forces, health officials said.
The identity and motive of the assailants were unclear. Aid workers have faced mistrust in some areas as they work to contain an Ebola outbreak.
Dozens of armed militia also regularly attack civilians and security forces in eastern Congo's borderlands with Uganda and Rwanda, which has significantly hampered the response to the disease.
The health ministry said in a statement that 38 suspected Ebola patients and 12 confirmed cases were in the center at the time of the attack. Four of the patients with confirmed cases fled and are being looked for, it said.
None of the patients who have been accounted for were injured, nor were any staff members, the ministry added.
French medical charity Medecins Sans Frontieres (MSF), which runs the center together with the ministry, condemned the "deplorable attack" and said its efforts were focused on the immediate safety of patients and staff.
The attack in the city of Butembo was the second in Congo's Ebola-hit east this week. On Sunday unidentified assailants set fire to a treatment center in the nearby town of Katwa, killing a nurse.
The current Ebola outbreak, first declared last August, is the second deadliest of the haemorrhagic fever since it was discovered in Congo in 1976. It is believed to have killed at least 553 people so far and infected over 300 more.
Reporting by Fiston Mahamba; Additional reporting by Giulia Paravicini; Writing by Aaron Ross; Editing by Gareth Jones and Rosalba O'Brien
FDA finds 'significant violations' at India's Strides plant
(Reuters) – The U.S. Food and.
U.N. wants more urgency in AIDS fight as gains and funding fade
LONDON (Reuters) – The global fight.
Oklahoma seeks to hold J&J responsible for opioid crisis as trial ends
NORMAN, Okla. (Reuters) – Lawyers for.
LONDON (Reuters) – Living healthily with.
Democrats take aim as Trump abandons drug pricing plan
WASHINGTON (Reuters) – Democrats see U.S..
U.S. appeals court blocks Trump administration birth control exemptions
(Reuters) – A federal appeals court.
China vows to tackle dead pig scam amid swine fever epidemic
FILE PHOTO: Piglets are seen by.
Alberta joins Ontario in pulling some CannTrust weed products
TORONTO (Reuters) – Alberta is placing.
Trump administration pushes U.S. at-home kidney care, transplant availability
NEW YORK/WASHINGTON (Reuters) – The Trump.
U.S. appeals court signals sympathy to bid to strike down Obamacare
NEW ORLEANS (Reuters) – A federal.
U.S. judge strikes down Trump administration rule requiring drug prices in TV ads
(Reuters) – A federal judge on.
Intra-Cellular drug fails one of two bipolar depression studies; shares fall
(Reuters) – Intra-Cellular Therapies Inc said.
China reports new African swine fever outbreak in Guangxi region
BEIJING (Reuters) – China's southwestern region.
Weed ban means no Rocky Mountain high for Canada's Calgary Stampede
CALGARY, Alberta (Reuters) – Canada's Calgary.
Weightlifting better at reducing heart fat than aerobic exercise
Obese people who engaged in resistance.
Sydney's city council reviews use of Bayer's Roundup weed killer amid cancer fears
SYDNEY (Reuters) – Sydney's city council.
Bulgaria confirms two cases of African swine fever
SOFIA (Reuters) – Bulgaria has confirmed.
FDA approves Karyopharm Therapeutics' blood cancer drug
Facebook to tackle content with misleading health claims
FILE PHOTO: The Facebook logo is.
FDA ties three deaths to Edwards Lifesciences' recalled heart devices
India asks its states not to partner with Philip Morris-funded foundation
NEW DELHI (Reuters) – India's health.
U.S. federal court delays adoption of healthcare rule on abortion
FILE PHOTO – A imaging table.
China tightens vaccine management after scandals
FILE PHOTO – A nurse prepares.
U.S wages lost to unpaid family care to hit $147 billion by 2050
As the U.S. population ages, the.
U.S. 'gag rule' linked to 40% jump in abortions in parts of Africa
LONDON (Reuters) – A decades-old U.S..
Study shows major real-world impact of cervical cancer vaccines
LONDON, June 26 (Reuters) – Vaccination.
Glyphosate use will eventually end, Merkel says
German Chancellor Angela Merkel attends a.
Unproven stem cell therapies often peddled by doctors without training
At U.S. clinics advertising unproven stem.
Teleflex recalls breathing tubes worldwide after reports of 4 deaths
(Reuters) – Teleflex Inc said on.
In border camps, Syrians rely on doctors in trucks and tents
AZAZ, Syria (Reuters) – The Syrian.
Thailand bans pig imports from Laos after African swine fever outbreak
BANGKOK (Reuters) – Thailand will ban.
FDA approves drug for loss of sexual desire in women
(Reuters) – The U.S. drug regulator.
Man trapped inside pair of skinny fit jeans
The New York Knicks Continue To Fill Their Big Baby Diapers With Poo Over NY Daily News Reports


Hazardous materials and waste are part of many work situations and can be found on many types of job sites. Make sure your facilities comply with OSHA's HAZWOPER regulation on hazardous cleanup with this course. This microlearning course is designed to explain the regulations and to help employees begin to reduce or eliminate potential exposure to hazardous materials in their work environments. Life-saving topics covered: the hazard communication program, the emergency response plan, the five levels of HAZWOPER emergency response training, personal protective equipment, site characterization, spill cleanup, decontamination procedures, and more.


Can Paint Color Inspire an Offer or Turn Potential Buyers Away?
Home | Remodeling, Selling Advice | Can Paint Color Inspire an Offer or Turn Potential Buyers Away?
Although it is often recommended that you add a fresh coat of paint when listing your home for sale, the color that you decide on may end up having an effect on potential purchasers' reactions. This goes for both the interior and the exterior of the property.
For example, the outside of your home is the first thing that people will see when they approach. And, just like with first impressions in other situations, an initial conclusion will usually be made by a potential buyer within just a minute or so.
That being the case, it is typically better to stick with a neutral color. A tan, gray, or white shade can translate into positive feelings, such as safety or shelter, while going with more of a brown hue can internalize warmth. In some cases, even a more bold color like red or blue could work. For instance, a lighter shade of blue can be universally appealing to many lookers, as can red – but only in smaller doses.
On the other hand, some colors can be a turn off to those who are viewing a home's curb appeal. In this case, pink, orange, purple, and yellow are normally not going to work – unless you happen to attract a potential purchaser who is very specific about what they are looking for.
Once a possible buyer has entered the home, the interior colors can be equally important. Here, according to Zillow.com, a kitchen that has an off white or creamy yellow will typically fare the best. Likewise, sticking with "earthy" tones on the rest of the walls, such as gray or even sage green, can oftentimes reel in an offer better than darker shades can.
If you're considering putting your home on the market, Contact Us. We can provide you with additional information – as well as give you an analysis of your home's present value.


Univision jumped into the short-form comedy game with THE FLAMA, a digital-first publisher highlighting all aspects of Latino and Hispanic culture among young people. They were looking for more Colombian focused content, and, since my wife is a Colombian actress and producer, we collaborated on a series of videos that put the spotlight on weird, interesting, or just fun aspects of Colombian culture.
Whether it's clingy roommates, a dirth of hot sauce, or the dreaded gringo dance meltdown, there's a video for that.
Not every loco DJ out there is so loco she can't give a sound lit-crit response.
I will make this into a movie one day. I will.


The global transcritical CO2 market is anticipated to reach USD 22.44 billion by 2026 according to a new study published by Polaris Market Research.
Increasing application of transcritical CO2 in food processing industry as refrigerant is the major factor driving the product demand. It is used as a substitute for environmentally harmful refrigerants including HCFCs, CFCs, and HFCs. The development of transcritical CO2 as a refrigerant has shown potential in many applications regarding system cost and efficiency both and makes it viable for several applications that is not considered feasible. Some of the mechanism where transcritical CO2 is currently used includes heat pumps water heaters for commercial, domestic and industrial applications, refrigeration systems in supermarkets, chest freezers of ice creams, beverage coolers, water chillers for moderate climates industrial and air conditioning and refrigeration in trains and busses. The other under development and less commercialized systems that make use of the product include mobile heat pumps, transport refrigeration systems in containers and ships, reversible and space heating in residential heat pumps, vending machines for combined cold and hot beverages, residential and non-residential combined cooling and heating buildings, heat pumps for combined space, water heating and space heating. The requirements of the product in increasing number of application systems are expected to be the major factor driving its demand from several industries.
In comparison to the traditional refrigerants, transcritical carbon dioxide has the best remarkable property of having low critical temperature of 31.1 degree Celsius. Owing to this, it leads to three discrete features of carbon dioxide in the refrigeration systems. These include that heat is rejected at supercritical pressures, keeps the pressure level quite high and during heat rejection it causes gliding of large refrigerant temperatures. These fundamental functionality advantages on application of transcritical carbon dioxide are expected to help in higher penetration of its applications sectors. Hence, with such advantages of the product and increasing application segments, the market value of TC CO2 is expected to increase over the forecast period.
North America was the leading regional market for consumption of transcritical CO2 refrigeration systems. U.S. was the largest contributor to product demand in 2017. Regulations concerning the limitations of refrigerants use and leakages along with directions for using TC CO2 refrigerants in supermarkets are expected to drive further product demand. The U.S. is also a leading manufacturer and exporter of heat pumps, which is yet another potential factor driving the industry. Asia Pacific is expected to be another leading player in the global scenario. The expanding regional machinery manufacturing base and growing TC CO2 enabled cooling systems is anticipated to boost product demand.
Some of leading and currently active industry participants include Shecco SPRL, Mayekawa MSG. Co., Ltd., Systems LMP, Inc., Henry Technologies, Inc. , Baltimore Aircoil Company, Inc. , Advansor A/S, Danfoss GmbH, Bitzer Kühlmaschinenbau GmbH , SCM Frigo SPA, Emerson Climate Technologies, Carnot Refrigeration Inc., Green & Cool World Refrigeration Ab, Carrier Commercial Refrigeration, and Hill Phoenix Inc.


We are Scotland's First Environmentally Friendly Launderette open 7 days a week in the Grassmarket, Edinburgh.
We offer a self service as well as service wash, dry and fold for customers. Those wishing for an ironing service can also be accommodated.
Laundry and pressing is done on-site and we accept both domestic and commercial work.
Other services include dry cleaning, wet cleaning, shoe cobbling, minor repairs and alterations, curtain cleaning, duvet cleaning and carpet cleaning.


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Accenture and UniSA join forces to develop the digital business skills of tomorrow
Accenture and UniSA are joining forces to offer a Bachelor of Digital Business.
Leading global professional services company Accenture, and the University of South Australia are today announcing a unique joint venture to help transform the provision of business education and digital skills in Australia.
The Innovation Academy in Digital Business will combine the best of Accenture's global practice and expertise with UniSA's proven leadership in transformative, industry-informed business education and research.
Kicking off in 2022, The Innovation Academy will deliver bespoke innovative curriculum and training programs, drawing on the skills and expertise of Accenture's global workforce and UniSA's established position in online education. Together the partnership will help address known and emerging skills gaps and enhance productivity in Australia's strategic growth areas.
The Academy's first offering - the Bachelor of Digital Business program, was designed and custom built leveraging real-world Accenture material and case studies. The second offering of the Academy focuses on the upskilling and reskilling of established workforces in leading-edge digital business practices.
UniSA Vice Chancellor Professor David Lloyd says: "UniSA is delighted to be working with Accenture to bring the Innovation Academy to life. This will be a global powerhouse in educating a new breed of business professional, informed by best practice and insight from one of the world's leading technology and strategic consulting companies. Our students will be gaining invaluable skills, and will be not just studying with the best, they'll be learning from the best as well."
Focused on the intersection between technology and business, the professional development component of the Academy will offer real-time, academically-moderated, practical upskilling programs featuring tailored digital business training modules. Built around UniSA's innovative Q-Credit assessment system, Academy learners will graduate with recognised tertiary qualifications, and highly employable and in-demand credentials.
Jordan Griffiths, Accenture Operations Lead Australia & New Zealand says: "We are thrilled to be working with UniSA and believe the Innovation Academy will transform how we think about digital business education, locally and globally. The Academy will give students real-world applicability to their studies as well as contributing to the upskilling of Australia, by developing talent that is equipped for the technology-driven future."
Enrolment for the new online Bachelor of Digital Business program will be live from mid-2022.
For more information and to register your interest in the program, go to: https://unisa.edu.au/connect/corporatepartners/accenture/
Accenture is a global professional services company with leading capabilities in digital, cloud and security. Combining unmatched experience and specialised skills across more than 40 industries, we offer Strategy and Consulting, Interactive, Technology and Operations services — all powered by the world's largest network of Advanced Technology and Intelligent Operations centers. Our 569,000 people deliver on the promise of technology and human ingenuity every day, serving clients in more than 120 countries. We embrace the power of change to create value and shared success for our clients, people, shareholders, partners and communities. Visit us at www.accenture.com.
About the University of South Australia
The University of South Australia is a globally connected university that partners with more than 2500 companies and industries worldwide. We are also South Australia's largest university with more than 37,000 students located across six campuses. Our teaching is industry-informed, our research is inventive and adventurous. We create knowledge that is central to global economic and social prosperity. Our graduates are the new professionals, global citizens at ease with the world and ready to create and respond to change. We are Australia's University of Enterprise. Visit us at www.unisa.edu.au
Georgia Hewett: 0417 699 567 georgia.hewett@accenture.com
Bronwyn Hurrell: 0427 393 656 bronwyn.hurrell@unisa.edu.au
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But not just Cedrus deodara, but Cedrus deodara glauca pendula. (That means a Weeping Blue Christmas tree,) Lucky you!
I currently have the tree in a training pot and it seems to be doing well.
For now I just want it to grow as it pleases, and hopefully fill out the empty mid-section.


i'm flying home today from a long trip to nyc. with so many friends in town, the mood was extra festive. it was cold as all get out, but i had the right gear for what felt like the first time, so there was lots of exploring, eating and enjoying the holiday city vibes.
loved staying in the LES – inspiration everywhere. probably my favorite hood.
it's all we need, right?
and last night we were lucky enough to score tickets to hamilton, and got to visit backstage after the show!


Christopher Stefani of Lawrence Tech talks education and industry.
Christopher Stefani, Associate Director of Lawrence Technological University's Detroit Center for Design and Technology sat with us to explore the steps the school is taking towards the transformative models in STEAM—Science, Technology, Engineering, Art, and Mathematics— and the role they are playing in it.
Detroitisit: What does STEAM represent for you and LTU?
STEAM for us is just that, areas of study that run both parallel and perpendicular to one another, but most importantly this will be the place where new tech and markets emerge from where they intersect.
DII: The ages of 12-14 that you're targeting for the camp are prime for pliable minds to be veering into so many different directions, how do you propose getting them to focus.
CS: I think, as in the case with most adolescents, there are a few keys to engaging the youth and honing their energy into focused work for the task set before them.
First, we're cultivating a relationship with the students similar to the way that any successful leadership does with their fellowship. Creating a mutual respect and peer-like relationship allows the youth to get invested in their work and to feel ownership over not only the results but the process.
Second, we're honing in on their interests and making a clear connection between how the smaller activity is plugging into the larger project. All the students who are attending this camp have an underlying interest in design and entrepreneurship – but many times the small tasks that are assigned feel like busy work. The goal here is to get every student invested in every part of the process and show them the value in designing the end result; logo to spread sheet. Everything deserves focus detailed work.
Finally, and very importantly, we're creating a healthy work culture. Similar to the way that many organizations are now working, the camp is set up with long, intensive work periods and fun activities that promote movement and creativity throughout.
DII: The program proposes some lofty goals to be accomplished in five days – develop a business plan, build a prototype, and more – do you believe the students will not only walk with this knowledge but know to think back on how to apply it for their future goals? And how do you propose to assist in that process?
CS: We know that no one is going to leave here as an expert on design thinking and business creation – but why not? We are looking to make this process and information intuitive, relatable and rational so that the students are able to grasp the concepts and then practice them.
Yes, we believe students will understand how to apply these concepts. It comes back to LTU's original mantra of Theory and Practice – once you have a working ability to apply knowledge through practice, you are able to apply the theory to different situations.
DII: What do you think it is about the practice of design that can get those outside of or new to the field thinking about solutions through design in a new way?
CS: The idea of maybe that is inherent to the practice of design is key – maybe yes, maybe no. After all, anything is possible right?
Design is a means of problem solving. Where we differ is the ability to look at creating solutions that shift the status quo, reject it, or even accept it and add to it. I believe everything is possible in design until you prove it can't be done – and this type of mindset breeds innovation.
DII: Why are you working with the partners you have for the program? What do they offer that others may not?
CS: I really admire the work of the Henry Ford Learning Institute. They are very passionate about their work with students and they have a clear and concise program outline for the Ford STEAM Lab camps. Their past participants have produced excellent results and there is a shared belief between our two institutions when it comes to the ideals of youth economic empowerment and how we educate students in the K-12 pipeline.
With regards to our generous partnership / sponsorship of .Design Domains, this was a no brainer! They flawlessly host the Center's website and really are forward-thinking in regard to branding in the digital age for the design industry. All of our design teams for the workshop will be pitching their products through the launch of a .design website.
DII: What's your dream to be accomplished by this year's program?
CS: We aim to the students' eyes to the possibilities that exist for them in design. We want to help them realize their potential and curate a goal – or plan – or pathway for a child to look toward, to chase. I think students have such amazing potential. They are truly a wealth of knowledge and fresh perspectives; untarnished by the realities we face in adulthood. When we invest in them, and help them understand the value in investing in themselves, we are creating a better future. We're working toward a more resilient and educated citizenry, who will be the leaders of tomorrow.


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As a former New York Times contributor I am wondering why you all have missed the ball on reporting about the environment? About Climate Change? Why is there is no environmental desk at your magazine? No Climate Watch?
This despite the fact that the US Military, an overwhelming number of scientists, the US President and many prime ministers throughout the world as well as leaders like the Pope and now billions of people see climate change as the greatest threat that humanity has ever faced.
It seems that your only real competitor, the UK Guardian, has left you in the proverbial radioactive fracking dust here.
I write this not as a science writer, photographer and journalist, but as a human being out here on the frontline of what is a huge calamity, unfolding in slow motion. You all are there, in New York City, also on the frontline, facing the fastest rising seas in the world. Your former Mayor and now the most recent one who marched through streets that four years ago were flooded by hurricane Sandy say, "Its Climate Change, Stupid".
These events will overtake us. They are overtaking us. According to your very own Thomas Friedman, the Syrian conflict started because of climate change related problems. Islands are being lost, people are losing their countries, their homes, their lives. Humanity must come together now.
From the pages of your newspaper one would imagine that its just another one of many stories, hardly mentionable. There on the science page we see how to not catch a cold on an airplane, a dry piece about how Global warming MAY make the California drought worse- no mention of how much water California uses for extreme energy like fracking. There is a lead article on how pretty the Earth is from space, an article on methane reduction in natural gas production that reads like an industry press release with no mention at all of the three recently published articles in the Journal of Geophysical Research out of the University of Colorado that have wildly different results than the Washington State study.
This stuff looks hand picked to appease industry, gloss over problems and is not meeting climate change head on. Where is the outcry and cutting stories over the crazed science deniers that have taken over our Congress? Why are you not going after them, rooting out their funders, digging into oil and gas money warping our politics?
We all depend upon you.
We all need to do our part but what part is that? You all have a role and responsibility to engage with this huge issue, give us real data and options to understand it and be a real leader.
Otherwise you all are part of the problem.
This entry was posted in Editorial, Environment. Bookmark the permalink.
You are correct. the NYT shrinking print news-hole shows a dearth of environmental reporting, but more than enough on the latest Republican stooge to jump into the presidential race, and full sections on fashion, the horror show that is NYC real estate, celebrity chefs. No analysis of the Asian takeover of the African continent, the trafficking of pangolins, lion and tiger parts, rhino assaults, ivory. Global warming gets attention, especially as relates to the local urban coastline. Its ad-driven.


text:
\section{Introduction}
\label{section: introduction}

An analysis of binary hypothesis testing from an
information-theoretic point of view, and a derivation of its
related error exponents in analogy to optimum channel codes was
provided in \cite{Blahut_IT74}. A nice exposition of the subject
is also provided in \cite[Chapter~11]{Cover and Thomas} where the
exact error exponents for the large deviation analysis of binary
hypothesis testing are provided in terms of relative entropies.

The Azuma-Hoeffding inequality is by now a well-known methodology
that has been often used to prove concentration of measure
phenomena. It is due to Hoeffding \cite{Hoeffding} who proved it
first for a sum of independent and bounded RVs, and Azuma
\cite{Azuma} who later extended it to bounded-difference
martingales. For a nice exposition of the martingale approach,
used for establishing concentration inequalities, the reader is
referred to e.g. \cite{survey2006} and \cite{McDiarmid_tutorial}.
The starting point of this work is an introduction of a known
concentration inequality for discrete-parameter martingales with
uniformly bounded jumps, which forms a refined version of the
Azuma-Hoeffding inequality. It is then used to study some of its
information-theoretic implications in the context of binary
hypothesis testing. Specifically, the tightness of this
concentration inequality is studied via a large deviations
analysis for binary hypothesis testing, and the demonstration of
its improved tightness over the Azuma-Hoeffding inequality is
revisited in this context. Some links of the derived lower bounds
on the error exponents to some information measures (e.g., the
relative entropy and Fisher information) are obtained along the
way.

This paper is structured as follows: Section~\ref{section:
Concentration Inequalities via Martingales} introduces briefly
some preliminary material related to martingales and Azuma's
inequality, and then it considers a refined version of Azuma's
inequality. This refined inequality is followed by a study of some
of its relation to the martingale central limit theorem.
Section~\ref{section: binary hypothesis testing} considers the
relation of the Azuma's inequality and the refined version of this
inequality (which was introduced in Section~\ref{section:
Concentration Inequalities via Martingales}) to large, moderate
and small deviations analysis of binary hypothesis testing.
Section~\ref{section: summary} concludes the paper, followed by
some proofs and complementary details that are relegated to the
appendices.

\section{Preliminaries and a New Concentration Inequality}
\label{section: Concentration Inequalities via Martingales}

In the following, we present briefly essential background on the
martingale approach that is used in this paper to derive
concentration inequalities. A refined version of Azuma's
inequality is then introduced. This concentration inequality is
applied in the next section for revising the large deviations
analysis of binary hypothesis testing.

\subsection{Doob's Martingales} \label{subsection: Martingales}
This sub-section provides a short background on martingales to set
definitions and notation. For a more thorough study of
martingales, the reader it referred to, e.g., \cite{Billingsley}.

\begin{definition}{\bf[Doob's Martingale]} Let $(\Omega, \mathcal{F},
\pr)$ be a probability space. A Doob's martingale sequence is a
sequence $X_0, X_1, \ldots$ of random variables (RVs) and
corresponding sub $\sigma$-algebras $\mathcal{F}_0, \mathcal{F}_1,
\ldots$ (also denoted by $\{X_i, \mathcal{F}_i\}$) that satisfy
the following conditions:
\begin{enumerate}
\item $X_i \in \LL^1(\Omega, \mathcal{F}_i, \pr)$ for every $i$, i.e., each $X_i$
is defined on the same sample space $\Omega$, it is measurable
with respect to the corresponding $\sigma$-algebra $\mathcal{F}_i$
(i.e., $X_i$ is $\mathcal{F}_i$-measurable) and $\expectation
[|X_i|] = \int_{\Omega} |X_i(\omega)| d\pr(\omega) < \infty.$
\item $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \ldots $ (where this sequence
of $\sigma$-algebras is called a filtration).
\item $X_i = \expectation[ X_{i+1} | \mathcal{F}_i]$
holds almost surely (a.s.) for every $i$.
\end{enumerate}
\label{definition: Doob's martingales}
\end{definition}

For preliminary material on the construction of discrete-time
martingales, see Appendix~\ref{Appendix: remarks on Doob's
martingales} (which is relevant to the analysis in
Section~\ref{section: binary hypothesis testing}).

\subsection{Azuma's Inequality} \label{subsection:
Azuma's inequality} Azuma's inequality\footnote{Azuma's inequality
is also known as the Azuma-Hoeffding inequality. Since this
inequality is referred several times in this paper, it will be
named from this point as Azuma's inequality for the sake of
brevity.} forms a useful concentration inequality for
bounded-difference martingales \cite{Azuma}. In the following,
this inequality is introduced. The reader is referred to, e.g.,
\cite[Chapter~11]{AlonS_tpm3}, \cite{survey2006} and
\cite{McDiarmid_tutorial} for surveys on concentration
inequalities for (sub/ super) martingales.

\begin{theorem}{\bf[Azuma's inequality]}
Let $\{X_k, \mathcal{F}_k\}_{k=0}^{\infty}$ be a
discrete-parameter real-valued martingale sequence such that for
every $k \in \naturals$, the condition $ |X_k - X_{k-1}| \leq d_k$
holds a.s. for some non-negative constants
$\{d_k\}_{k=1}^{\infty}$. Then
\begin{equation}
\pr( | X_n - X_0 | \geq r) \leq 2 \exp\left(-\frac{r^2}{2
\sum_{k=1}^n d_k^2}\right) \, \quad \forall \, r \geq 0.
\label{eq: Azuma's concentration inequality - general case}
\end{equation}
\label{theorem: Azuma's concentration inequality}
\end{theorem}

The concentration inequality stated in Theorem~\ref{theorem:
Azuma's concentration inequality} was proved in \cite{Hoeffding}
for independent bounded random variables, followed by a discussion
on sums of dependent random variables; this inequality was later
derived in \cite{Azuma} for bounded-difference martingales. For a
proof of Theorem~\ref{theorem: Azuma's concentration inequality}
see, e.g., \cite{survey2006} and
\cite[Chapter~2.4]{Dembo_Zeitouni}.

\subsection{A Refined Version of Azuma's Inequality}

\begin{theorem}
Let $\{X_k, \mathcal{F}_k\}_{k=0}^{\infty}$ be a
discrete-parameter real-valued martingale. Assume that, for some
constants $d, \sigma > 0$, the following two requirements are
satisfied a.s.
\begin{eqnarray*}
&& | X_k - X_{k-1} | \leq d, \\
&& \text{Var} (X_k | \mathcal{F}_{k-1}) = \expectation \bigl[(X_k
- X_{k-1})^2 \, | \, \mathcal{F}_{k-1} \bigr] \leq \sigma^2
\end{eqnarray*}
for every $k \in \{1, \ldots, n\}$. Then, for every $\alpha \geq
0$,
\begin{equation}
\hspace*{-0.2cm} \pr(|X_n-X_0| \geq \alpha n) \leq 2 \exp\left(-n
\, D\biggl(\frac{\delta+\gamma}{1+\gamma} \Big|\Big|
\frac{\gamma}{1+\gamma}\biggr) \right) \label{eq: first refined
concentration inequality}
\end{equation}
where
\begin{equation}
\gamma \triangleq \frac{\sigma^2}{d^2}, \quad \delta \triangleq
\frac{\alpha}{d}  \label{eq: notation}
\end{equation}
and
\begin{equation}
D(p || q) \triangleq p \ln\Bigl(\frac{p}{q}\Bigr) + (1-p)
\ln\Bigl(\frac{1-p}{1-q}\Bigr), \quad \forall \, p, q \in [0,1]
\label{eq: divergence}
\end{equation}
is the divergence (a.k.a. relative entropy or Kullback-Leibler
distance) between the two probability distributions $(p,1-p)$ and
$(q,1-q)$. If $\delta>1$, then the probability on the left-hand
side of \eqref{eq: first refined concentration inequality} is
equal to zero. \label{theorem: first refined concentration
inequality}
\end{theorem}
\begin{proof}
The idea of the proof of Theorem~\ref{theorem: first refined
concentration inequality} is essentially similar to the proof of
\cite[Corollary~2.4.7]{Dembo_Zeitouni}. The full proof is provided
in \cite[Section~III]{Sason_submitted_paper}.
\end{proof}

\begin{proposition}
Let $\{X_k, \mathcal{F}_k\}_{k=0}^{\infty}$ be a
discrete-parameter real-valued martingale. Then, for every $\alpha
\geq 0$,
\begin{equation}
\pr(|X_n-X_0| \geq \alpha \sqrt{n}) \leq 2
\exp\Bigl(-\frac{\delta^2}{2\gamma}\Bigr) \Bigl(1+
O\bigl(n^{-\frac{1}{2}}\bigr)\Bigr). \label{eq: concentration1}
\end{equation}
\label{proposition: a similar scaling of the concentration
inequalities}
\end{proposition}
\begin{proof}
This inequality follows from Theorem~\ref{theorem: first refined
concentration inequality} (see
\cite[Appendix~G]{Sason_submitted_paper}).
\end{proof}

\section{Binary Hypothesis Testing}
\label{section: binary hypothesis testing}

Binary hypothesis testing for finite alphabet models was analyzed
via the method of types, e.g., in \cite[Chapter~11]{Cover and
Thomas} and \cite{Csiszar_Shields_FnT}. It is assumed that the
data sequence is of a fixed length $(n)$, and one wishes to make
the optimal decision (based on the Neyman-Pearson ratio test)
based on the received sequence.

Let the RVs $X_1, X_2 ....$ be i.i.d. $\sim Q$, and consider two
hypotheses:
\begin{itemize}
\item $H_1:  Q = P_1$.
\item $H_2:  Q = P_2$.
\end{itemize}
For the simplicity of the analysis, let us assume that the RVs are
discrete, and take their values on a finite alphabet $\mathcal{X}$
where $P_1(x), P_2(x) > 0$ for every $x \in \mathcal{X}$.

In the following, let
\begin{equation*}
L(X_1, \ldots, X_n) \triangleq \ln \frac{P_1^n(X_1, \ldots,
X_n)}{P_2^n(X_1, \ldots, X_n)} = \sum_{i=1}^n \ln
\frac{P_1(X_i)}{P_2(X_i)}
\end{equation*}
designate the log-likelihood ratio. By the strong law of large
numbers (SLLN), if hypothesis $H_1$ is true, then a.s.
\begin{equation}
\lim_{n \rightarrow \infty} \frac{L(X_1, \ldots, X_n)}{n} = D(P_1
|| P_2) \label{eq: a.s. limit of the normalized LLR under
hypothesis H1}
\end{equation}
and otherwise, if hypothesis $H_2$ is true, then a.s.
\begin{equation}
\lim_{n \rightarrow \infty} \frac{L(X_1, \ldots, X_n)}{n} = -D(P_2
|| P_1) \label{eq: a.s. limit of the normalized LLR under
hypothesis H2}
\end{equation}
where the above assumptions on the probability mass functions
$P_1$ and $P_2$ imply that the relative entropies, $D(P_1 || P_2)$
and $D(P_2 || P_1)$, are both finite. Consider the case where for
some fixed constants $\overline{\lambda}, \underline{\lambda} \in
\reals$ where $$-D(P_2||P_1) < \underline{\lambda} \leq
\overline{\lambda} < D(P_1||P_2)$$ one decides on hypothesis $H_1$
if $$ L(X_1, \ldots, X_n) > n \overline{\lambda} $$ and on
hypothesis $H_2$ if $$ L(X_1, \ldots, X_n) < n
\underline{\lambda}.$$ Note that if $\overline{\lambda} =
\underline{\lambda} \triangleq \lambda$ then a decision on the two
hypotheses is based on comparing the normalized log-likelihood
ratio (w.r.t. $n$) to a single threshold $(\lambda)$, and deciding
on hypothesis $H_1$ or $H_2$ if this normalized log-likelihood
ratio is, respectively, above or below $\lambda$. If
$\underline{\lambda} < \overline{\lambda}$ then one decides on
$H_1$ or $H_2$ if the normalized log-likelihood ratio is,
respectively, above the upper threshold $\overline{\lambda}$ or
below the lower threshold $\underline{\lambda}$. Otherwise, if the
normalized log-likelihood ratio is between the upper and lower
thresholds, then an erasure is declared and no decision is taken
in this case.

Let
\begin{eqnarray}
&& \alpha_n^{(1)} \triangleq P_1^n \Bigl( L(X_1, \ldots, X_n) \leq
n \overline{\lambda} \Bigr)
\label{eq: error and erasure event under hypothesis H1} \\
&& \alpha_n^{(2)} \triangleq P_1^n \Bigl( L(X_1, \ldots, X_n) \leq
n \underline{\lambda} \Bigr) \label{eq: error event under
hypothesis H1}
\end{eqnarray}
and
\begin{eqnarray}
&& \beta_n^{(1)}  \triangleq P_2^n \Bigl( L(X_1, \ldots, X_n) \geq
n \underline{\lambda} \Bigr)
\label{eq: error and erasure event under hypothesis H2} \\
&& \beta_n^{(2)}  \triangleq P_2^n \Bigl( L(X_1, \ldots, X_n) \geq
n \overline{\lambda} \Bigr) \label{eq: error event under
hypothesis H2}
\end{eqnarray}
then $\alpha_n^{(1)}$ and $\beta_n^{(1)}$ are the probabilities of
either making an error or declaring an erasure under,
respectively, hypotheses $H_1$ and $H_2$; similarly
$\alpha_n^{(2)}$ and $\beta_n^{(2)}$ are the probabilities of
making an error under hypotheses $H_1$ and $H_2$, respectively.

Let $\pi_1, \pi_2 \in (0,1)$ denote the a-priori probabilities of
the hypotheses $H_1$ and $H_2$, respectively, so
\begin{equation}
P_{\text{e}, n}^{(1)} = \pi_1 \alpha_n^{(1)} + \pi_2 \beta_n^{(1)}
\label{eq: overall probability of a mixed error and erasure event}
\end{equation}
is the probability of having either an error or an erasure, and
\begin{equation}
P_{\text{e}, n}^{(2)} = \pi_1 \alpha_n^{(2)} + \pi_2 \beta_n^{(2)}
\label{eq: overall error probability}
\end{equation}
is the probability of error.

\subsection{Exact Exponents} When we let $n$ tend to infinity,
the exact exponents of $\alpha_n^{(j)}$ and $\beta_n^{(j)}$
($j=1,2$) are derived via Cram\'{e}r's theorem. The resulting
exponents form a straightforward generalization of, e.g.,
\cite[Theorem~3.4.3]{Dembo_Zeitouni} and
\cite[Theorem~6.4]{Hollander_book_2000} that addresses the case
where the decision is made based on a single threshold of the
log-likelihood ratio. In this particular case where
$\overline{\lambda} = \underline{\lambda} \triangleq \lambda$, the
option of erasures does not exist, and $P_{\text{e}, n}^{(1)} =
P_{\text{e}, n}^{(2)} \triangleq P_{\text{e}, n}$ is the error
probability.

In the considered general case with erasures, let $$ \lambda_1
\triangleq -\overline{\lambda}, \quad \lambda_2 \triangleq
-\underline{\lambda} $$ then Cram\'{e}r's theorem on $\reals$
yields that the exact exponents of $\alpha_n^{(1)}$,
$\alpha_n^{(2)}$, $\beta_n^{(1)}$ and $\beta_n^{(2)}$ are given by
\begin{eqnarray}
&& \lim_{n \rightarrow \infty} -\frac{\ln \alpha_n^{(1)}}{n} =
I(\lambda_{1})
\label{eq: exponent of alpha_n1} \\[0.1cm]
&& \lim_{n \rightarrow \infty} -\frac{\ln \alpha_n^{(2)}}{n} =
I(\lambda_{2})
\label{eq: exponent of alpha_n2} \\[0.1cm]
&& \lim_{n \rightarrow \infty} -\frac{\ln \beta_n^{(1)}}{n} =
I(\lambda_2) - \lambda_2 \label{eq: exponent of beta_n1} \\[0.1cm]
&& \lim_{n \rightarrow \infty} -\frac{\ln \beta_n^{(2)}}{n} =
I(\lambda_1) - \lambda_1 \label{eq: exponent of beta_n2}
\end{eqnarray}
where the rate function $I$ is given by
\begin{equation}
I(r) \triangleq \sup_{t \in \reals} \bigl( tr - H(t) \bigr)
\label{eq: rate function}
\end{equation}
and
\begin{equation}
H(t) = \ln \Biggl(\sum_{x \in \mathcal{X}} P_1(x)^{1-t} P_2(x)^t
\Biggr), \quad \forall \, t \in \reals. \label{eq: H}
\end{equation}
The rate function $I$ is convex, lower semi-continuous (l.s.c.)
and non-negative (see, e.g., \cite{Dembo_Zeitouni} and
\cite{Hollander_book_2000}). Note that
$$H(t) = (t-1) D_t(P_2||P_1)$$ where $D_t(P||Q)$ designates R\'{e}yni's
information divergence of order $t$, and $I$ in \eqref{eq: rate
function} is the Fenchel-Legendre transform of $H$ (see, e.g.,
\cite[Definition~2.2.2]{Dembo_Zeitouni}).

From \eqref{eq: overall probability of a mixed error and erasure
event}-- \eqref{eq: exponent of beta_n2}, the exact exponents of
$P_{\text{e}, n}^{(1)}$ and $P_{\text{e}, n}^{(2)}$ are equal to
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}^{(1)}}{n}
= \min \Bigl\{ I(\lambda_1), I(\lambda_2) - \lambda_2 \Bigr\}
\label{eq: exact exponent of the overall error and erasure
probability}
\end{equation}
and
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}^{(2)}}{n}
= \min \Bigl\{ I(\lambda_2), I(\lambda_1) - \lambda_1 \Bigr\}.
\label{eq: exact error exponent}
\end{equation}

For the case where the decision is based on a single threshold for
the log-likelihood ratio (i.e., $\lambda_1 = \lambda_2 \triangleq
\lambda$), then $P_{\text{e}, n}^{(1)} = P_{\text{e}, n}^{(2)}
\triangleq P_{\text{e}, n}$, and its error exponent is equal to
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}}{n} = \min
\Bigl\{ I(\lambda), I(\lambda) - \lambda \Bigr\} \label{eq: exact
exponent of the error probability for a single threshold}
\end{equation}
which coincides with the error exponent in
\cite[Theorem~3.4.3]{Dembo_Zeitouni} (or
\cite[Theorem~6.4]{Hollander_book_2000}). The optimal threshold
for obtaining the best error exponent of the error probability
$P_{\text{e}, n}$ is equal to zero (i.e., $\lambda=0$); in this
case, the exact error exponent is equal to
\begin{equation}
\hspace*{-0.2cm} I(0) = -\min_{0 \leq t \leq 1} \ln \Biggl(
\sum_{x \in \mathcal{X}} P_1(x)^{1-t} P_2(x)^t \Biggr) \triangleq
C(P_1, P_2) \label{eq: Chernoff information}
\end{equation}
which is the Chernoff information of the probability measures
$P_1$ and $P_2$ (see \cite[Eq.~(11.239)]{Cover and Thomas}), and
it is symmetric (i.e., $C(P_1, P_2) = C(P_2, P_1)$). Note that,
from \eqref{eq: rate function}, $I(0) = \sup_{t \in
\reals}\bigl(-H(t)\bigr) = -\inf_{t \in \reals}\bigl(H(t)\bigr)$;
the minimization in \eqref{eq: Chernoff information} over the
interval $[0,1]$ (instead of taking the infimum of $H$ over
$\reals$) is due to the fact that $H(0) = H(1) = 0$ and the
function $H$ in \eqref{eq: H} is convex, so it is enough to
restrict the infimum of $H$ to the closed interval $[0,1]$ for
which it turns to be a minimum.

\subsection{Lower Bound on the Exponents via
Theorem~\ref{theorem: first refined concentration inequality}} In
the following, the tightness of Theorem~\ref{theorem: first
refined concentration inequality} is examined by using it for the
derivation of lower bounds on the error exponent and the exponent
of the event of having either an error or an erasure. These
results will be compared in the next sub-section to the exact
exponents from the previous sub-section.

We first derive a lower bound on the exponent of $\alpha_n^{(1)}$.
Under hypothesis $H_1$, let us construct the martingale sequence
$\{U_k, \mathcal{F}_k\}_{k=0}^n$ where $\mathcal{F}_0 \subseteq
\mathcal{F}_1 \subseteq \ldots \mathcal{F}_n$ is the filtration
$$ \mathcal{F}_0 = \{\emptyset, \Omega\}, \quad \mathcal{F}_k =
\sigma(X_1, \ldots, X_k), \; \; \forall \, k \in \{1, \ldots,
n\}$$ and
\begin{equation}
U_k = \expectation_{P_1^n} \bigl[ L(X_1, \ldots, X_n) \; | \;
\mathcal{F}_k \bigr]. \label{eq: martingale sequence U under
hypothesis H1}
\end{equation}
For every $k \in \{0, \ldots, n\}$
\begin{eqnarray*}
&& U_k = \expectation_{P_1^n} \Biggl[  \sum_{i=1}^n
\ln \frac{P_1(X_i)}{P_2(X_i)} \; \Big| \; \mathcal{F}_k  \Biggr] \\
&& \hspace*{0.5cm} =  \sum_{i=1}^k \ln \frac{P_1(X_i)}{P_2(X_i)} +
\sum_{i=k+1}^n \expectation_{P_1^n} \Biggl[
\ln \frac{P_1(X_i)}{P_2(X_i)} \Biggr] \\
&& \hspace*{0.5cm} =  \sum_{i=1}^k \ln \frac{P_1(X_i)}{P_2(X_i)} +
(n-k) D(P_1 || P_2).
\end{eqnarray*}
In particular
\begin{eqnarray}
&& U_0 = n D(P_1 || P_2), \label{eq: initial value of the
martingale U that is related to the binary hypothesis testing}   \\
&& U_n = \sum_{i=1}^n \ln \frac{P_1(X_i)}{P_2(X_i)} = L(X_1,
\ldots, X_n) \label{eq: final value of the martingale U that is
related to the binary hypothesis testing}
\end{eqnarray}
and, for every $k \in \{1, \ldots, n\}$,
\begin{equation}
U_k - U_{k-1} = \ln \frac{P_1(X_k)}{P_2(X_k)} - D(P_1 || P_2).
\label{eq: jumps of the martingale U that is related to the binary
hypothesis testing}
\end{equation}
Let
\begin{equation}
d_1 \triangleq \max_{x \in \mathcal{X}} \left| \ln
\frac{P_1(x)}{P_2(x)} - D(P_1 || P_2) \right| \label{eq: d1}
\end{equation}
so $d_1 < \infty$ since by assumption the alphabet set
$\mathcal{X}$ is finite, and $P_1(x), P_2(x) > 0$ for every $x \in
\mathcal{X}$. From \eqref{eq: jumps of the martingale U that is
related to the binary hypothesis testing} and \eqref{eq: d1}
$$|U_k - U_{k-1}| \leq d_1$$ holds a.s. for every $k \in \{1, \ldots,
n\}$, and
\begin{eqnarray}
&& \expectation_{P_1^n} \bigl[ (U_k - U_{k-1})^2 \, | \,
\mathcal{F}_{k-1} \bigr] \nonumber \\
&& = \expectation_{P_1} \left[ \left( \ln
\frac{P_1(X_k)}{P_2(X_k)}
- D(P_1 || P_2) \right)^2 \right] \nonumber \\
&& = \sum_{x \in \mathcal{X}} \left\{ P_1(x) \left( \ln
\frac{P_1(x)}{P_2(x)} - D(P_1 || P_2) \right)^2
\right\} \nonumber \\
&& \triangleq \sigma_1^2. \label{eq: sigma1 squared for the jumps
of the martingale U}
\end{eqnarray}

Let
\begin{eqnarray}
&& \hspace*{-1cm} \varepsilon_{1,1} = D(P_1 || P_2) -
\overline{\lambda}, \quad \varepsilon_{2,1} = D(P_2 || P_1) +
\underline{\lambda}
\label{eq: the epsilons introduced for mixed errors and erasures in binary hypothesis testing} \\
&& \hspace*{-1cm} \varepsilon_{1,2} = D(P_1 || P_2) -
\underline{\lambda}, \quad \varepsilon_{2,2} = D(P_2 || P_1) +
\overline{\lambda} \label{eq: the epsilons introduced for errors
in binary hypothesis testing}
\end{eqnarray}
The probability of making an erroneous decision on hypothesis
$H_2$ or declaring an erasure under the hypothesis $H_1$ is equal
to $\alpha_n^{(1)}$, and from Theorem~\ref{theorem: first refined
concentration inequality}
\begin{eqnarray}
&& \alpha_n^{(1)} \triangleq P_1^n \bigl( L(X_1, \ldots, X_n)
\leq n \overline{\lambda} \bigr) \nonumber \\
&& \hspace*{0.7cm} \stackrel{\text{(a)}}{=} P_1^n(U_n - U_0 \leq
-\varepsilon_{1,1} \, n)
\label{eq: intermediate step in the derivation of a bound on alpha1} \\
&& \hspace*{0.7cm} \stackrel{\text{(b)}}{\leq} \exp \left(-n \,
D\Bigl(\frac{\delta_{1,1} + \gamma_1}{1+\gamma_1} \Big|\Big|
\frac{\gamma_1}{1+\gamma_1} \Bigr) \right) \label{eq:
concentration inequality for the first error event}
\end{eqnarray}
where equality~(a) follows from \eqref{eq: initial value of the
martingale U that is related to the binary hypothesis testing},
\eqref{eq: final value of the martingale U that is related to the
binary hypothesis testing} and \eqref{eq: the epsilons introduced
for mixed errors and erasures in binary hypothesis testing}, and
inequality~(b) follows from Theorem~\ref{theorem: first refined
concentration inequality} with
\begin{equation}
\gamma_1 \triangleq \frac{\sigma_1^2}{d_1^2}, \quad \delta_{1,1}
\triangleq \frac{\varepsilon_{1,1}}{d_1}. \label{eq: gamma1 and
delta1,1}
\end{equation}
Note that if $\varepsilon_{1,1} > d_1$ then it follows from
\eqref{eq: jumps of the martingale U that is related to the binary
hypothesis testing} and \eqref{eq: d1} that $\alpha_n^{(1)}$ is
zero; in this case $\delta_{1,1} > 1$, so the divergence in
\eqref{eq: concentration inequality for the first error event} is
infinity and the upper bound is also equal to zero. Hence, it is
assumed without loss of generality that $\delta_{1,1} \in [0, 1]$.

Similarly to \eqref{eq: martingale sequence U under hypothesis
H1}, under hypothesis~$H_2$, let us define the martingale sequence
$\{U_k, \mathcal{F}_k\}_{k=0}^n$ with the same filtration and
\begin{equation}
\hspace*{-0.4cm} U_k = \expectation_{P_2^n} \bigl[ L(X_1, \ldots,
X_n) \; | \; \mathcal{F}_k \bigr], \quad \forall \, k \in \{0,
\ldots, n\}. \label{eq: martingale sequence U  under hypothesis
H2}
\end{equation}
For every $k \in \{0, \ldots, n\}$
\begin{eqnarray*}
&& U_k = \sum_{i=1}^k \ln \frac{P_1(X_i)}{P_2(X_i)} - (n-k) D(P_2
|| P_1)
\end{eqnarray*}
and in particular
\begin{equation}
U_0 = -n D(P_2 || P_1), \quad U_n = L(X_1, \ldots, X_n).
\label{eq: initial and final values of the second martingale
sequence}
\end{equation}
For every $k \in \{1, \ldots, n\}$,
\begin{equation}
U_k - U_{k-1} = \ln \frac{P_1(X_k)}{P_2(X_k)} + D(P_2 || P_1).
\label{eq: jumps of the martingale U under hypothesis H2}
\end{equation}
Let
\begin{equation}
d_2 \triangleq \max_{x \in \mathcal{X}} \left| \ln
\frac{P_2(x)}{P_1(x)} - D(P_2 || P_1) \right| \label{eq: d2}
\end{equation}
then, the jumps of the latter martingale sequence are uniformly
bounded by $d_2$ and, similarly to \eqref{eq: sigma1 squared for
the jumps of the martingale U}, for every $k \in \{1, \ldots, n\}$
\begin{eqnarray}
&& \expectation_{P_2^n} \bigl[ (U_k - U_{k-1})^2 \, | \,
\mathcal{F}_{k-1} \bigr] \nonumber \\
&& = \sum_{x \in \mathcal{X}} \left\{ P_2(x) \left( \ln
\frac{P_2(x)}{P_1(x)} - D(P_2 || P_1) \right)^2 \right\} \nonumber
\\ && \triangleq \sigma_2^2. \label{eq: sigma2 squared for the
jumps of the martingale U}
\end{eqnarray}
Hence, it follows from Theorem~\ref{theorem: first refined
concentration inequality} that
\begin{eqnarray}
&& \beta_n^{(1)} \triangleq P_2^n \bigl( L(X_1, \ldots, X_n)
\geq n \underline{\lambda} \bigr) \nonumber \\
&& \hspace*{0.7cm} = P_2^n(U_n - U_0 \geq \varepsilon_{2,1} \, n)
\label{eq: intermediate step in the derivation of a bound on beta1} \\
&& \hspace*{0.7cm} \leq  \exp \left(-n \,
D\Bigl(\frac{\delta_{2,1} + \gamma_2}{1+\gamma_2} \Big|\Big|
\frac{\gamma_2}{1+\gamma_2} \Bigr) \right) \label{eq:
concentration inequality for the second error event}
\end{eqnarray}
where the equality in \eqref{eq: intermediate step in the
derivation of a bound on beta1} holds due to \eqref{eq: initial
and final values of the second martingale sequence} and \eqref{eq:
the epsilons introduced for mixed errors and erasures in binary
hypothesis testing}, and \eqref{eq: concentration inequality for
the second error event} follows from Theorem~\ref{theorem: first
refined concentration inequality} with
\begin{equation}
\gamma_2 \triangleq \frac{\sigma_2^2}{d_2^2}, \quad \delta_{2,1}
\triangleq \frac{\varepsilon_{2,1}}{d_2} \label{eq: gamma2 and
delta2,1}
\end{equation}
and $d_2$, $\sigma_2$ are introduced, respectively, in \eqref{eq:
d2} and \eqref{eq: sigma2 squared for the jumps of the martingale
U}.

From \eqref{eq: overall probability of a mixed error and erasure
event}, \eqref{eq: concentration inequality for the first error
event} and \eqref{eq: concentration inequality for the second
error event}, the exponent of the probability of either having an
error or an erasure is lower bounded by
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}^{(1)}}{n}
\geq \min_{i=1,2} D\Bigl(\frac{\delta_{i,1} +
\gamma_i}{1+\gamma_i} \Big|\Big| \frac{\gamma_i}{1+\gamma_i}
\Bigr). \label{eq: lower bound on the exponent of mixed errors and
erasures for binary hypothesis testing}
\end{equation}
Similarly to the above analysis, one gets from \eqref{eq: overall
error probability} and \eqref{eq: the epsilons introduced for
errors in binary hypothesis testing} that the error exponent is
lower bounded by
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}^{(2)}}{n}
\geq \min_{i=1,2} D\Bigl(\frac{\delta_{i,2} +
\gamma_i}{1+\gamma_i} \Big|\Big| \frac{\gamma_i}{1+\gamma_i}
\Bigr) \label{eq: lower bound on the error exponent for binary
hypothesis testing}
\end{equation}
where
\begin{equation}
\delta_{1,2} \triangleq \frac{\varepsilon_{1,2}}{d_1}, \quad
\delta_{2,2} \triangleq \frac{\varepsilon_{2,2}}{d_2}. \label{eq:
delta1,2 and delta2,2}
\end{equation}

For the case of a single threshold (i.e., $\overline{\lambda} =
\underline{\lambda} \triangleq \lambda$) then \eqref{eq: lower
bound on the exponent of mixed errors and erasures for binary
hypothesis testing} and \eqref{eq: lower bound on the error
exponent for binary hypothesis testing} coincide, and one obtains
that the error exponent satisfies
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}}{n} \geq
\min_{i=1,2} D\Bigl(\frac{\delta_i + \gamma_i}{1+\gamma_i}
\Big|\Big| \frac{\gamma_i}{1+\gamma_i} \Bigr) \label{eq: lower
bound on the error exponent for binary hypothesis testing with a
single threshold}
\end{equation}
where $\delta_i$ is the common value of $\delta_{i,1}$ and
$\delta_{i,2}$ (for $i=1,2$). In this special case, the zero
threshold is optimal (see, e.g., \cite[p.~93]{Dembo_Zeitouni}),
which then yields that \eqref{eq: lower bound on the error
exponent for binary hypothesis testing with a single threshold} is
satisfied with
\begin{equation}
\delta_1 = \frac{D(P_1 || P_2)}{d_1}, \quad \delta_2 = \frac{D(P_2
|| P_1)}{d_2} \label{eq: delta1 and delta2}
\end{equation}
with $d_1$ and $d_2$ from \eqref{eq: d1} and \eqref{eq: d2},
respectively. The right-hand side of \eqref{eq: lower bound on the
error exponent for binary hypothesis testing with a single
threshold} forms a lower bound on Chernoff information which is
the exact error exponent for this special case.

\subsection{Comparison of the Lower Bounds on the Exponents with
those that Follow from Azuma's Inequality} The lower bounds on the
error exponent and the exponent of the probability of having
either errors or erasures, that were derived in the previous
sub-section via Theorem~\ref{theorem: first refined concentration
inequality}, are compared in the following to the loosened lower
bounds on these exponents that follow from Azuma's inequality.

We first obtain upper bounds on $\alpha_n^{(1)}, \alpha_n^{(2)},
\beta_n^{(1)}$ and $\beta_n^{(2)}$ via Azuma's inequality, and
then use them to derive lower bounds on the exponents of
$P_{\text{e},n}^{(1)}$ and $P_{\text{e},n}^{(2)}$.

From \eqref{eq: jumps of the martingale U that is related to the
binary hypothesis testing}, \eqref{eq: d1}, \eqref{eq:
intermediate step in the derivation of a bound on alpha1},
\eqref{eq: gamma1 and delta1,1}, and Azuma's inequality
\begin{equation}
\alpha_n^{(1)} \leq \exp \biggl(-\frac{\delta_{1,1}^2 n}{2}
\biggr) \label{eq: Azuma's inequality for alpha1}
\end{equation}
and, similarly, from \eqref{eq: jumps of the martingale U under
hypothesis H2}, \eqref{eq: d2}, \eqref{eq: intermediate step in
the derivation of a bound on beta1}, \eqref{eq: gamma2 and
delta2,1}, and Azuma's inequality
\begin{equation}
\beta_n^{(1)} \leq \exp \biggl(-\frac{\delta_{2,1}^2 n}{2}
\biggr). \label{eq: Azuma's inequality for beta1}
\end{equation}
From \eqref{eq: error event under hypothesis H1}, \eqref{eq: error
event under hypothesis H2}, \eqref{eq: the epsilons introduced for
errors in binary hypothesis testing}, \eqref{eq: delta1,2 and
delta2,2} and Azuma's inequality
\begin{eqnarray}
&& \hspace*{-0.5cm} \alpha_n^{(2)} \leq \exp
\biggl(-\frac{\delta_{1,2}^2 n}{2} \biggr)
\label{eq: Azuma's inequality for alpha2} \\
&& \hspace*{-0.5cm} \beta_n^{(2)} \leq \exp
\biggl(-\frac{\delta_{2,2}^2 n}{2} \biggr). \label{eq: Azuma's
inequality for beta2}
\end{eqnarray}
Therefore, it follows from \eqref{eq: overall probability of a
mixed error and erasure event}, \eqref{eq: overall error
probability} and \eqref{eq: Azuma's inequality for
alpha1}--\eqref{eq: Azuma's inequality for beta2} that the
resulting lower bounds on the exponents of $P_{\text{e},n}^{(1)}$
and $P_{\text{e},n}^{(2)}$ are
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}^{(j)}}{n}
\geq \min_{i=1,2} \frac{\delta_{i,j}^2}{2}, \quad j = 1, 2
\label{eq: lower bounds on the exponents for binary hypothesis
testing via Azuma inequality}
\end{equation}
as compared to \eqref{eq: lower bound on the exponent of mixed
errors and erasures for binary hypothesis testing} and \eqref{eq:
lower bound on the error exponent for binary hypothesis testing}
which give, for $j=1,2$,
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}^{(j)}}{n}
\geq \min_{i=1,2} D\Bigl(\frac{\delta_{i,j} +
\gamma_i}{1+\gamma_i} \Big|\Big| \frac{\gamma_i}{1+\gamma_i}
\Bigr). \label{eq: lower bounds on the exponents for binary
hypothesis testing via Theorem 2}
\end{equation}
For the specific case of a zero threshold, the lower bound on the
error exponent which follows from Azuma's inequality is given by
\begin{equation}
\lim_{n \rightarrow \infty} - \frac{\ln P_{\text{e}, n}^{(j)}}{n}
\geq \min_{i=1,2} \frac{\delta_i^2}{2} \label{eq: loosened lower
bound on the error exponent for zero threshold}
\end{equation}
with the values of $\delta_1$ and $\delta_2$ in \eqref{eq: delta1
and delta2}.

The lower bounds on the exponents in \eqref{eq: lower bounds on
the exponents for binary hypothesis testing via Azuma inequality}
and \eqref{eq: lower bounds on the exponents for binary hypothesis
testing via Theorem 2} are compared in the following. Note that
the lower bounds in \eqref{eq: lower bounds on the exponents for
binary hypothesis testing via Azuma inequality} are loosened as
compared to those in \eqref{eq: lower bounds on the exponents for
binary hypothesis testing via Theorem 2} since they follow,
respectively, from Azuma's inequality and its improvement in
Theorem~\ref{theorem: first refined concentration inequality}.

The divergence in the exponent of \eqref{eq: lower bounds on the
exponents for binary hypothesis testing via Theorem 2} is equal to

\small \vspace*{-0.2cm}
\begin{eqnarray}
&& \hspace*{-0.8cm} D\Bigl(\frac{\delta_{i,j} +
\gamma_i}{1+\gamma_i}
\Big|\Big| \frac{\gamma_i}{1+\gamma_i} \Bigr) \nonumber \\[0.1cm]
&& \hspace*{-0.8cm} =
\left(\frac{\delta_{i,j}+\gamma_i}{1+\gamma_i} \right) \ln \left(
1 + \frac{\delta_{i,j}}{\gamma_i} \right) +
\left(\frac{1-\delta_{i,j}}{1+\gamma_i}\right)
\ln(1-\delta_{i,j}) \nonumber \\[0.1cm]
&& \hspace*{-0.8cm} = \frac{\gamma_i}{1+\gamma_i} \left[ \left( 1
+ \frac{\delta_{i,j}}{\gamma_i} \right) \ln \Bigl( 1 +
\frac{\delta_{i,j}}{\gamma_i}
\Bigr) + \frac{(1-\delta_{i,j}) \ln(1-\delta_{i,j})}{\gamma_i}\right]. \nonumber \\
\label{eq: equality for the divergence}
\end{eqnarray}

\normalsize
\begin{lemma}
\begin{equation}
(1+u) \ln(1+u) \geq \left\{
\begin{array}{ll}
u + \frac{u^2}{2}, \quad & u \in [-1, 0] \\[0.2cm]
u+\frac{u^2}{2}-\frac{u^3}{6}, \quad & u \geq 0
\end{array}
\right. \label{eq: inequality for lower bounding the divergence}
\end{equation}
where at $u=-1$, the left-hand side is defined to be zero (it is
the limit of this function when $u \rightarrow -1$ from above).
\label{lemma: inequality for lower bounding the divergence}
\end{lemma}
\begin{proof}
The proof follows by elementary calculus.
\end{proof}

Since $\delta_{i,j} \in [0,1]$, then \eqref{eq: equality for the
divergence} and Lemma~\ref{lemma: inequality for lower bounding
the divergence} imply that
\begin{equation}
D\Bigl(\frac{\delta_{i,j} + \gamma_i}{1+\gamma_i} \Big|\Big|
\frac{\gamma_i}{1+\gamma_i} \Bigr) \geq \frac{\delta_{i,j}^2}{2
\gamma_i} - \frac{\delta_{i,j}^3}{6 \gamma_i^2 (1+\gamma_i)}.
\label{eq: lower bound on the lower bound of the exponents}
\end{equation}
Hence, by comparing \eqref{eq: lower bounds on the exponents for
binary hypothesis testing via Azuma inequality} with the
combination of \eqref{eq: lower bounds on the exponents for binary
hypothesis testing via Theorem 2} and \eqref{eq: lower bound on
the lower bound of the exponents}, then it follows that (up to a
second-order approximation) the lower bounds on the exponents that
were derived via Theorem~\ref{theorem: first refined concentration
inequality} are improved by at least a factor of $\bigl(\max
\gamma_i\bigr)^{-1}$ as compared to those that follow from Azuma's
inequality.

\begin{example}
Consider two probability measures $P_1$ and $P_2$ where
$$P_1(0) = P_2(1) = 0.4, \quad P_1(1) = P_2(0) = 0.6,$$
and the case of a single threshold of the log-likelihood ratio
that is set to zero (i.e., $\lambda = 0$). The exact error
exponent in this case is Chernoff information that is equal to
$$C(P_1, P_2) = 2.04 \cdot 10^{-2}.$$ The improved lower bound on
the error exponent in \eqref{eq: lower bound on the error exponent
for binary hypothesis testing with a single threshold} and
\eqref{eq: delta1 and delta2} is equal to $1.77 \cdot 10^{-2}$,
whereas the loosened lower bound in \eqref{eq: loosened lower
bound on the error exponent for zero threshold} is equal to $1.39
\cdot 10^{-2}$. In this case $\gamma_1 = \frac{2}{3}$ and
$\gamma_2 = \frac{7}{9}$, so the improvement in the lower bound on
the error exponent is indeed by a factor of approximately
$\left(\max_i \gamma_i \right)^{-1} = \frac{9}{7}.$ Note that,
from \eqref{eq: concentration inequality for the first error
event}, \eqref{eq: concentration inequality for the second error
event} and \eqref{eq: Azuma's inequality for alpha1}--\eqref{eq:
Azuma's inequality for beta2}, these are lower bounds on the error
exponents for any finite block length $n$, and not only
asymptotically in the limit where $n \rightarrow \infty$. The
operational meaning of this example is that the improved lower
bound on the error exponent assures that a fixed error probability
can be obtained based on a sequence of i.i.d. RVs whose length is
reduced by 22.2\% as compared to the loosened bound which follows
from Azuma's inequality.
\end{example}

\subsection{Comparison of the Exact and Lower Bounds on the Error
Exponents, Followed by a Relation to Fisher Information} In the
following, we compare the exact and lower bounds on the error
exponents. Consider the case where there is a single threshold on
the log-likelihood ratio (i.e., referring to the case where the
erasure option is not provided) that is set to zero. The exact
error exponent in this case is given by the Chernoff information
(see \eqref{eq: Chernoff information}), and it will be compared to
the two lower bounds on the error exponents that were derived in
the previous two subsections.

Let $\{P_{\theta}\}_{\theta \in \Theta}$, denote an indexed family
of probability mass functions where $\Theta$ denotes the parameter
set. Assume that $P_{\theta}$ is differentiable in the parameter
$\theta$. Then, the Fisher information is defined as
\begin{equation}
J(\theta) \triangleq \expectation_{\theta} \left[
\frac{\partial}{\partial \theta} \, \ln P_{\theta}(x) \right]^2
\label{eq: Fisher information}
\end{equation}
where the expectation is w.r.t. the probability mass function
$P_{\theta}$. The divergence and Fisher information are two
related information measures, satisfying the equality
\begin{equation}
\lim_{\theta' \rightarrow \theta} \frac{D(P_{\theta} ||
P_{\theta'})}{(\theta - \theta')^2} = \frac{J(\theta)}{2}
\label{eq: relation between the divergence and Fisher information}
\end{equation}
(note that if it was a relative entropy to base~2 then the
right-hand side of \eqref{eq: relation between the divergence and
Fisher information} would have been divided by $\ln 2$, and be
equal to $\frac{J(\theta)}{\ln 4}$ as in \cite[Eq.~(12.364)]{Cover
and Thomas}).
\begin{proposition}
Under the above assumptions,
\begin{itemize}
\item The Chernoff information and Fisher information are related
information measures that satisfy the equality
\begin{equation}
\lim_{\theta' \rightarrow \theta} \frac{C(P_{\theta},
P_{\theta'})}{(\theta - \theta')^2} = \frac{J(\theta)}{8}.
\label{eq: relation between the Chernoff information and Fisher
information}
\end{equation}
\item Let
\begin{equation}
E_{\text{L}}(P_{\theta}, P_{\theta'}) \triangleq \min_{i=1,2}
D\Bigl(\frac{\delta_i + \gamma_i}{1+\gamma_i} \Big|\Big|
\frac{\gamma_i}{1+\gamma_i} \Bigr) \label{eq: E_L}
\end{equation}
be the lower bound on the error exponent in \eqref{eq: lower bound
on the error exponent for binary hypothesis testing with a single
threshold} which corresponds to $P_1 \triangleq P_{\theta}$ and
$P_2 \triangleq P_{\theta'}$, then also
\begin{equation}
\lim_{\theta' \rightarrow \theta} \frac{E_{\text{L}}(P_{\theta},
P_{\theta'})}{(\theta - \theta')^2} = \frac{J(\theta)}{8}.
\label{eq: relation between the improved lower bound on the error
exponent and Fisher information}
\end{equation}
\item Let
\begin{equation}
\widetilde{E}_{\text{L}}(P_{\theta}, P_{\theta'}) \triangleq
\min_{i=1,2} \frac{\delta_i^2}{2} \label{eq: tilde E_L}
\end{equation}
be the loosened lower bound on the error exponent in \eqref{eq:
loosened lower bound on the error exponent for zero threshold}
which refers to $P_1 \triangleq P_{\theta}$ and $P_2 \triangleq
P_{\theta'}$. Then,
\begin{equation}
\lim_{\theta' \rightarrow \theta}
\frac{\widetilde{E}_{\text{L}}(P_{\theta}, P_{\theta'})}{(\theta -
\theta')^2} = \frac{a(\theta) \, J(\theta)}{8} \label{eq: relation
between the loosened lower bound on the error exponent and Fisher
information}
\end{equation}
for some deterministic function $a$ bounded in $[0, 1]$, and there
exists an indexed family of probability mass functions for which
$a(\theta)$ can be made arbitrarily close to zero for any fixed
value of $\theta \in \Theta$.
\end{itemize}
\label{proposition: Fisher information}
\end{proposition}
\begin{proof}
See Appendix~\ref{appendix: Fisher information}.
\end{proof}

\vspace*{0.1cm} Proposition~\ref{proposition: Fisher information}
shows that, in the considered setting, the refined lower bound on
the error exponent provides the correct behavior of the error
exponent for a binary hypothesis testing when the relative entropy
between the pair of probability mass functions that characterize
the two hypotheses tends to zero. This stays in contrast to the
loosened error exponent, which follows from Azuma's inequality,
whose scaling may differ significantly from the correct exponent
(for a concrete example, see the last part of the proof in
Appendix~\ref{appendix: Fisher information}).

\begin{example}
Consider the index family of of probability mass functions defined
over the binary alphabet $\mathcal{X} = \{0,1\}$:
$$ P_\theta(0) = 1-\theta, \; \; P_\theta(1) = \theta, \quad
\forall \, \theta \in (0,1).$$ From \eqref{eq: Fisher
information}, the Fisher information is equal to
$$ J(\theta) = \frac{1}{\theta} + \frac{1}{1-\theta}$$
and, at the point $\theta = 0.5$, $J(\theta) = 4$. Let $\theta_1 =
0.51$ and $\theta_2 = 0.49$, so from \eqref{eq: relation between
the Chernoff information and Fisher information} and \eqref{eq:
relation between the improved lower bound on the error exponent
and Fisher information}
$$ C(P_{\theta_1}, P_{\theta_2}), E_{\text{L}}(P_{\theta_1}, P_{\theta_2}) \approx
\frac{J(\theta) (\theta_1 - \theta_2)^2}{8} = 2.00 \cdot
10^{-4}.$$ Indeed, the exact values of $C(P_{\theta_1},
P_{\theta_2})$ and $E_{\text{L}}(P_{\theta_1}, P_{\theta_2})$ are
$2.000 \cdot 10^{-4}$ and $1.997 \cdot 10^{-4}$, respectively.
\end{example}

\section{Summary}
\label{section: summary} This work introduces a concentration
inequality for discrete-parameter martingales with uniformly
bounded jumps, which forms a refined version of Azuma's
inequality. The tightness of this concentration inequality is
studied via a large deviations analysis of binary hypothesis
testing, and the demonstration of its improved tightness over
Azuma's inequality is revisited in this context. Some links of the
derived lower bounds on the error exponents to some information
measures (e.g., the relative entropy and Fisher information) are
obtained along the way. This paper presents in part the work in
\cite{Sason_submitted_paper} where further concentration
inequalities that form a refinement of Azuma's inequality were
derived, followed by some further applications of these
concentration inequalities in information theory, communication,
and coding theory. It is meant to stimulate the use of some
refined versions of the Azuma-Hoeffding inequality in
information-theoretic aspects.

\appendices

\section{Some complementary remarks concerning the
construction of Doob's martingales} \label{Appendix: remarks on
Doob's martingales}

This appendix is relevant to the analysis in Section~\ref{section:
binary hypothesis testing}.

\begin{remark} Let $\{X_i, \mathcal{F}_i\}$ be a martingale sequence.

For every $i$, $\expectation[X_{i+1}] = \expectation \bigl[
\expectation[X_{i+1} | \mathcal{F}_i] \bigr] = \expectation[X_i]$,
so the expectation of a martingale stays constant.
\end{remark}

\begin{remark} One can generate martingale sequences by the
following procedure: Given a RV $X \in \LL^1(\Omega, \mathcal{F},
\pr)$ and an arbitrary filtration of sub $\sigma$-algebras
$\{\mathcal{F}_i\}$, let
\begin{equation*}
X_i = \expectation[X | \mathcal{F}_i] \quad i = 0, 1, \ldots .
\end{equation*}
Then, the sequence $X_0, X_1, \ldots$ forms a martingale since
\begin{enumerate}
\item The RV $X_i = \expectation[X |
\mathcal{F}_i]$ is $\mathcal{F}_i$-measurable, and also
$\expectation[|X_i|] \leq \expectation[|X|] < \infty$ (since
conditioning reduces the expectation of the absolute value).
\item By construction $\{\mathcal{F}_i\}$ is a filtration.
\item From the tower principle for conditional expectations,
since $\{\mathcal{F}_i\}$ is a filtration, then for every $i$
\begin{equation*}
\expectation[X_{i+1} | \mathcal{F}_i] = \expectation \bigl[
\expectation[ X | \mathcal{F}_{i+1}] | \mathcal{F}_i \bigr] =
\expectation[X | \mathcal{F}_i]  \quad \text{a.s}.
\end{equation*}
\end{enumerate}
\label{remark: construction of martingales}
\end{remark}

\begin{remark}
In continuation to Remark~\ref{remark: construction of
martingales}, one can choose $\mathcal{F}_0 = \{\Omega, \emptyset
\}$ and $\mathcal{F}_n = \mathcal{F}$. Hence, $X_0, X_1, \ldots,
X_n$ is a martingale sequence where
\begin{eqnarray*}
&& \hspace*{-0.8cm} X_0 = \expectation[X | \mathcal{F}_0] =
\expectation[X] \quad
\text{(since $X$ is independent of $\mathcal{F}_0$)} \\
&& \hspace*{-0.8cm} X_n = \expectation[X | \mathcal{F}_n] = X \;
\; \text{a.s.} \quad \text{(since $X$ is
$\mathcal{F}$-measurable)}.
\end{eqnarray*}
This has the following interpretation: At the beginning, we don't
know anything about $X$, so it is initially estimated by its
expectation. We then reveal at each step more and more information
about $X$ until we can specify it completely (a.s.).
\label{remark: construction of martingales (cont.)}
\end{remark}

\section{Proof of Proposition~\ref{proposition: Fisher information}}
\label{appendix: Fisher information} The proof of \eqref{eq:
relation between the Chernoff information and Fisher information}
is based on calculus, and it is similar to the proof of the limit
in \eqref{eq: relation between the divergence and Fisher
information} that relates the divergence and Fisher information.
For the proof of \eqref{eq: relation between the improved lower
bound on the error exponent and Fisher information}, note that

\vspace*{-0.3cm} \small
\begin{equation}
\hspace*{-0.25cm} C(P_{\theta}, P_{\theta'}) \geq
E_{\text{L}}(P_{\theta}, P_{\theta'}) \geq \min_{i=1,2}
\left\{\frac{\delta_i^2}{2 \gamma_i} - \frac{\delta_i^3}{6
\gamma_i^2 (1+\gamma_i)} \right\}. \label{eq: bounds on the
improved lower bound}
\end{equation}
\normalsize The left-hand side of \eqref{eq: bounds on the
improved lower bound} holds since $E_{\text{L}}$ is a lower bound
on the error exponent, and the exact value of this error exponent
is the Chernoff information. The right-hand side of \eqref{eq:
bounds on the improved lower bound} follows from Lemma~\ref{lemma:
inequality for lower bounding the divergence} (see \eqref{eq:
lower bound on the lower bound of the exponents}) and the
definition of $E_{\text{L}}$ in \eqref{eq: E_L}. By definition
$\gamma_i \triangleq \frac{\sigma_i^2}{d_i^2}$ and $\delta_i
\triangleq \frac{\varepsilon_i}{d_i}$ where, based on \eqref{eq:
delta1 and delta2},
\begin{equation}
\varepsilon_1 \triangleq D(P_\theta || P_{\theta'}), \quad
\varepsilon_2 \triangleq D(P_\theta' || P_\theta). \label{eq:
epsilon1,2}
\end{equation}
The term on the left-hand side of \eqref{eq: bounds on the
improved lower bound} therefore satisfies
\begin{eqnarray*}
&& \frac{\delta_i^2}{2 \gamma_i} -
\frac{\delta_i^3}{6 \gamma_i^2 (1+\gamma_i)} \\
&& = \frac{\varepsilon_i^2}{2 \sigma_i^2} - \frac{\varepsilon_i^3
d_i^3}{6 \sigma_i^2 (\sigma_i^2 + d_i^2)} \geq
\frac{\varepsilon_i^2}{2 \sigma_i^2} \left( 1 -
\frac{\varepsilon_i d_i}{3} \right)
\end{eqnarray*}
so it follows from \eqref{eq: bounds on the improved lower bound}
and the last inequality that
\begin{equation}
\hspace*{-0.25cm} C(P_{\theta}, P_{\theta'}) \geq
E_{\text{L}}(P_\theta, P_{\theta'}) \geq \min_{i=1,2} \left\{
\frac{\varepsilon_i^2}{2 \sigma_i^2} \left( 1 -
\frac{\varepsilon_i d_i}{3} \right) \right\}. \label{eq: 2nd ver.
for the bounds on the improved lower bound}
\end{equation}
Based on the continuity assumption of the indexed family
$\{P_{\theta}\}_{\theta \in \Theta}$, then it follows from
\eqref{eq: epsilon1,2} that
$$\lim_{\theta' \rightarrow \theta} \varepsilon_i=0,
\quad \forall \, i \in \{ 1, 2\}$$ and also, from \eqref{eq: d1}
and \eqref{eq: d2} with $P_1$ and $P_2$ replaced by $P_\theta$ and
$P_\theta'$ respectively, then
$$\lim_{\theta' \rightarrow \theta} d_i = 0, \quad \forall \, i \in \{ 1, 2\}.$$
It therefore follows from \eqref{eq: relation between the Chernoff
information and Fisher information} and \eqref{eq: 2nd ver. for
the bounds on the improved lower bound} that
\begin{equation}
\hspace*{-0.4cm} \frac{J(\theta)}{8} \geq \lim_{\theta'
\rightarrow \theta} \frac{E_{\text{L}}(P_{\theta},
P_{\theta'})}{(\theta - \theta')^2} \geq \lim_{\theta' \rightarrow
\theta} \, \min_{i=1,2} \left\{ \frac{\varepsilon_i^2}{2
\sigma_i^2 (\theta-\theta')^2} \right\}. \label{eq: bounds on the
limit}
\end{equation}
The idea is to show that the limit on the right-hand side of this
inequality is $\frac{J(\theta)}{8}$ (same as the left-hand side),
and hence, the limit of the middle term is also
$\frac{J(\theta)}{8}$.
\begin{eqnarray}
&& \hspace*{-0.7cm} \lim_{\theta' \rightarrow \theta}
\frac{\varepsilon_1^2}{2 \sigma_1^2 (\theta-\theta')^2}
\nonumber \\[0.1cm]
&& \hspace*{-0.7cm} \stackrel{(\text{a})}{=} \lim_{\theta'
\rightarrow \theta} \frac{D(P_\theta ||
P_{\theta'})^2}{2 \sigma_1^2 (\theta-\theta')^2} \nonumber \\[0.1cm]
&& \hspace*{-0.7cm} \stackrel{(\text{b})}{=} \frac{J(\theta)}{4}
\lim_{\theta' \rightarrow \theta} \frac{D(P_\theta ||
P_{\theta'})}{\sigma_1^2} \nonumber \\[0.1cm]
&& \hspace*{-0.7cm} \stackrel{(\text{c})}{=} \frac{J(\theta)}{4}
\lim_{\theta' \rightarrow \theta} \frac{D(P_\theta ||
P_{\theta'})}{\sum_{x \in \mathcal{X}} P_\theta(x) \left( \ln
\frac{P_\theta(x)}{P_{\theta'}(x)} - D(P_\theta ||
P_{\theta'}) \right)^2} \nonumber \\[0.1cm]
&& \hspace*{-0.7cm} \stackrel{(\text{d})}{=} \frac{J(\theta)}{4}
\lim_{\theta' \rightarrow \theta} \frac{D(P_\theta ||
P_{\theta'})}{\sum_{x \in \mathcal{X}} P_\theta(x) \left( \ln
\frac{P_\theta(x)}{P_{\theta'}(x)} \right)^2 \; - \; D(P_\theta ||
P_{\theta'})^2} \nonumber \\[0.1cm]
&& \hspace*{-0.7cm} \stackrel{(\text{e})}{=} \frac{J(\theta)^2}{8}
\lim_{\theta' \rightarrow \theta}
\frac{(\theta-\theta')^2}{\sum_{x \in \mathcal{X}} P_\theta(x)
\left( \ln \frac{P_\theta(x)}{P_{\theta'}(x)}
\right)^2 \; - \; D(P_\theta || P_{\theta'})^2} \nonumber \\[0.1cm]
&& \hspace*{-0.7cm} \stackrel{(\text{f})}{=} \frac{J(\theta)^2}{8}
\lim_{\theta' \rightarrow \theta}
\frac{(\theta-\theta')^2}{\sum_{x \in \mathcal{X}} P_\theta(x)
\left( \ln \frac{P_\theta(x)}{P_{\theta'}(x)}
\right)^2} \nonumber \\[0.1cm]
&& \hspace*{-0.7cm} \stackrel{(\text{g})}{=} \frac{J(\theta)}{8}
\label{eq: first calculated limit}
\end{eqnarray}
where equality~(a) follows from \eqref{eq: epsilon1,2},
equalities~(b), (e) and~(f) follow from \eqref{eq: relation
between the divergence and Fisher information},
equality~(c) follows from \eqref{eq: sigma1 squared for the jumps
of the martingale U} with $P_1 = P_{\theta}$ and $P_2 =
P_{\theta'}$, equality~(d) follows from the definition of the
divergence, and equality~(g) follows by calculus (the required
limit is calculated by using L'H\^{o}pital's rule twice) and from
the definition of Fisher information in \eqref{eq: Fisher
information}. Similarly, also
$$ \lim_{\theta' \rightarrow \theta} \frac{\varepsilon_2^2}{2
\sigma_2^2 (\theta-\theta')^2} = \frac{J(\theta)}{8}$$ so
$$ \lim_{\theta' \rightarrow \theta} \, \min_{i=1,2}
\left\{ \frac{\varepsilon_i^2}{2 \sigma_i^2 (\theta-\theta')^2}
\right\} = \frac{J(\theta)}{8}.$$ Hence, it follows from
\eqref{eq: bounds on the limit} that $ \lim_{\theta' \rightarrow
\theta} \frac{E_{\text{L}}(P_\theta, P_{\theta'})}{(\theta -
\theta')^2} = \frac{J(\theta)}{8}.$ This completes the proof of
\eqref{eq: relation between the improved lower bound on the error
exponent and Fisher information}.

We prove now Eq.~\eqref{eq: relation between the loosened lower
bound on the error exponent and Fisher information}. From
\eqref{eq: d1}, \eqref{eq: d2}, \eqref{eq: delta1 and delta2} and
\eqref{eq: tilde E_L}
\begin{equation*}
\widetilde{E}_{\text{L}}(P_\theta, P_{\theta'}) = \min_{i=1,2}
\frac{\varepsilon_i^2}{2 d_i^2}
\end{equation*}
with $\varepsilon_1$ and $\varepsilon_2$ in \eqref{eq:
epsilon1,2}. Hence,
\begin{equation*}
\lim_{\theta' \rightarrow \theta}
\frac{\widetilde{E}_{\text{L}}(P_\theta, P_{\theta'})}{(\theta' -
\theta)^2} \leq \lim_{\theta' \rightarrow \theta}
\frac{\varepsilon_1^2}{2 d_1^2 (\theta' - \theta)^2}
\end{equation*}
and from \eqref{eq: first calculated limit} and last inequality
then it follows that
\begin{eqnarray}
&& \hspace*{-1.2cm} \lim_{\theta' \rightarrow \theta}
\frac{\widetilde{E}_{\text{L}}(P_\theta, P_{\theta'})}{(\theta' - \theta)^2} \nonumber \\
&& \hspace*{-1.2cm} \leq \frac{J(\theta)}{8}
\lim_{\theta' \rightarrow \theta} \frac{\sigma_1^2}{d_1^2} \nonumber \\
&& \hspace*{-1.2cm} \stackrel{(\text{a})}{=} \frac{J(\theta)}{8}
\lim_{\theta' \rightarrow \theta} \frac{\sum_{x \in \mathcal{X}}
P_\theta(x) \left( \ln \frac{P_\theta(x)}{P_{\theta'}(x)} -
D(P_\theta || P_{\theta'}) \right)^2}{\biggl( \max_{x \in
\mathcal{X}} \left|  \ln \frac{P_\theta(x)}{P_{\theta'}(x)} -
D(P_\theta || P_{\theta'}) \right| \biggr)^2}. \label{eq: second
calculated limit}
\end{eqnarray}
It is clear that the second term on the right-hand side of
\eqref{eq: second calculated limit} is bounded between zero and
one (if the limit exists). This limit can be made arbitrarily
small, i.e., there exists an indexed family of probability mass
functions $\{P_{\theta}\}_{\theta \in \Theta}$ for which the
second term on the right-hand side of \eqref{eq: second calculated
limit} can be made arbitrarily close to zero. For a concrete
example, let $\alpha \in (0,1)$ be fixed, and $\theta \in
\reals^+$ be a parameter that defines the following indexed family
of probability mass functions over the ternary alphabet
$\mathcal{X} = \{0, 1, 2\}$: $$ P_{\theta}(0) =
\frac{\theta(1-\alpha)}{1+\theta}, \quad P_{\theta}(1) = \alpha,
\quad P_{\theta}(2) = \frac{1-\alpha}{1+\theta}. $$ Then, it
follows by calculus that for this indexed family
\begin{equation*}
\lim_{\theta' \rightarrow \theta} \frac{\sum_{x \in \mathcal{X}}
P_\theta(x) \left( \ln \frac{P_\theta(x)}{P_{\theta'}(x)} -
D(P_\theta || P_{\theta'}) \right)^2}{\biggl( \max_{x \in
\mathcal{X}} \left|  \ln \frac{P_\theta(x)}{P_{\theta'}(x)} -
D(P_\theta || P_{\theta'}) \right| \biggr)^2} = (1-\alpha) \theta
\end{equation*}
so, for any $\theta \in \reals^{+}$, the above limit can be made
arbitrarily close to zero by choosing $\alpha$ close enough to~1.
This completes the proof of \eqref{eq: relation between the
loosened lower bound on the error exponent and Fisher
information}, and also the proof of Proposition~\ref{proposition:
Fisher information}.

redpajama_set_name: RedPajamaArXiv


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corolla 12-20 mm wide at apex in life and mericarps erect, 5-9 mm tall (vs. A. officinalis, with the corolla 5-10 mm wide at apex in life and mericarps obliquely oriented, ca. 2 mm tall).
corolla zygomorphic, the basal tube bent, mericarps 1-2 mm tall, and plants annual (vs. A. officinalis, with the corolla actinomorphic, the basal tube straight, mericarps ca. 2 mm tall, and plants perennial).
common bugloss. Anchusa procera Bess. ex Link • CT, MA, ME, NH, RI. Roadsides, fields, dumps, disturbed soil.


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