DEATH WAS NEAR. They told her that. Chances were it could be weeks - perhaps longer but not significantly unless she had a lung transplant. For years, Ashley Owens had known that she would not live to be 30 or even 25, that cystic fibrosis would sweep her away one day before she would have a chance to have a career or a wedding or children. It was a given she had come to accept. But now that she was coughing up blood and was in what her doctors called the "the end stages," the sudden finality of her circumstances terrified her. All of it seemed to be happening too soon.
They told her that they would be moving her to the Hospital of the University of Pennsylvania. It was not something she wanted to do, if only because she had become accustomed to St. Christopher's Hospital for Children. She had been going there three or four times a year since she had been an 8-year-old and had befriended the nursing staff. But her doctors told her that there was a surgery that she would have to have, and that it was perhaps a good idea to become acquainted with the transplant team at Penn. Oddly, a feeling of calm settled over her at that point - what she would later describe as a "a trancelike state." So she found a pen and some paper and began writing goodbye letters: to her parents, Bob and Charlotte; her young brother, Robert; and her boyfriend, Jesse, the young man who stood by her through her worst days.
With a shaggy beard and gentle bearing, Jesse Quinter swept her off her feet, both figuratively and literally. When she had been too weak to walk somewhere, Jesse lifted her then 5-foot, 69-pound body up and carried her on his back. They had met each other in study hall at Owen J. Roberts High School in Pottstown. Ashley told him before their first date how sick she was, but he just shrugged and told her: "I like you for you." However worrisome her ordeal would become, Ashley would come to depend on Jesse to cheer her up. When she tearfully told him on the phone that day that she would be leaving for Penn, he left early from his job at the Warwick Child Care Center in Lionville and hurried to her side.
They talked. But she was upset and no words could seem to soothe her. Even when Jesse reassured her that she would be fine, she was in a forlorn place that seemed beyond even his reach. It was then that an idea popped into his head. He excused himself and said he had to get something from his car. When he came back, he sat down in a chair by her bed and resumed their conversation, which he always tried to keep light. Instead of dwelling on the sobering prognosis that faced her, Jesse would ask what she wanted to do when she got out of the hospital, where she would go to dinner and what trips she would like to take. It went on like that until he paused.
"I have to talk to you about something," he said.
Casually, Ashley replied: "About what?"
Jesse got down on one knee and displayed a diamond ring.
And with eyes wide, Ashley cried, "Oh, my God!
On the very evening this scene was unfolding last year - Friday, Nov. 20 - Francisco "Paco" Rodriguez was preparing to step into a boxing ring at the Blue Horizon, where he had a scheduled 12-round bout with Teon Kennedy for the vacant United States Boxing Association super bantamweight title. Paco was stopped by Kennedy in the 10th round, passed out in his corner and died of a head injury 2 days later at Hahnemann University Hospital. But it was there that one story ended and another began, the tale of how with a stroke of a pen on a consent form, a grieving widow bestowed life upon five people by offering seven organs from the body of her beloved husband for transplant donation. What began in a place of unutterable grief ended up in a realm of hope reborn.
Eighteen people die each day in the United States waiting for a transplant. In the case of the five people who received organs from Paco, each of their histories is tied together by a common thread: They had endured untold suffering in the grip of their various illnesses. Only days away from death in some cases, they looked upon themselves as fighters in the same very real sense that Paco had been. With the exception of his uncle, Ramon Tejeda, who received a kidney in a "direct donation," none of them had ever heard of the young boxer from Chicago. Given what they have received from him - a heart, a liver, two lungs, two kidneys and a pancreas - none of them will ever forget him. While the recipients have not yet met, they share a bond that now unites them with someone they had come to cherish: Paco.
* Ashley Owens, 23, of Spring City, Chester County: Both lungs.
As a 10-month-old baby, she weighed less than 7 pounds. Initially, doctors suspected she had a tumor. But tests revealed that she had cystic fibrosis, which compromised her breathing and to some extent her digestion. Simple childhood pleasures such as running and swimming were beyond her ability. In and out of the hospital during her school years, she became an excellent student with the help of a tutor. Physically, she began "going downhill" at age 20 or so, a period during which her lung capacity dropped to as low as 20 percent. Without the help of oxygen her lips would turn blue. Concerned by the statistics that foretold of an uncertain outcome for lung-transplant recipients, she held off going onto the waiting list until just hours before she suffered a collapsed lung on Nov. 13, 2009. Of the pain her daughter endured, Charlotte Owens says, "Some days she would push through it. Other days it would be more than she could bear."
Ashley says: "Until the last 2 or 3 years, I had an OK handle on it. But when I was 20, I had stopped responding to the medication I was taking. My body had become so full of it that I had become immune. They told me I had 2 years to live. When I was 21, they told me I had 1 year to live. I was scared."
* Meghan Kingsley, 26, of Gaithersburg, Md.: Liver.
At 16, she was diagnosed with neurofibromatosis type 2, characterized by the growth of non-cancerous tumors along the nerve that transmits information from the inner ear to the brain. An exceptional competitive swimmer who had dreamed one day of going to the Olympic Games, she underwent surgery in June 2001 for the removal of a tumor and was left deaf in one ear. In October 2007, she had decompression surgery on another tumor that doctors chose not to remove. In an effort to preserve what remained of her hearing, they instead carved away some bone that would allow the tumor room to grow. However, she began experiencing significant hearing loss and in September 2009 enrolled in a study for the experimental drug PCT299. By November, she was in the throes of liver failure.
Meghan says: "I became very, very ill and ended up in Johns Hopkins. I remember I was constantly burping; I had so much fluid in my stomach. I became jaundiced. [The whites of] my eyes were green and yellow. Mom said I looked like 'The Grinch.' I no longer had any bodily function. They later told me I was within 48 hours of dying."
* Alexis Sloan, 27, of Norristown: Heart.
At 22, she was diagnosed with congestive heart failure, prior to which she had experienced symptoms that included a dry cough, fatigue and shortness of breath. "A lot of big words were thrown at me," she says. "Scary." Within a year of her diagnosis, she received a biventricular pacemaker and defibrillator implant. Efforts to manage her condition with medication failed and in March 2007 she says she "coded," which is hospital slang for going into cardiopulmonary arrest. Doctors then equipped her with a left ventricular assist device (LVAD), which she found to be an unwieldy contraption. Battery-operated, it had internal and external components that left her feeling on some days as if she was a robot. To get on the waiting list for a heart, she had to fulfill a standard set of requirements that proved that she would submit to postoperative care. In May 2008, she had done that and was given a pager, with which she would be contacted when a heart was available.
Alexis says: "When they gave me the initial diagnosis, it was devastating. It seemed like a death sentence. There was a lot of confusion. When I got the LVAD, I was not happy with it. No young person should have to live that way. With the protocols I had to go through, it seemed like it was taking forever to get on the list. I became depressed and one point even suicidal. I just thought: 'I am going to die anyway . . . ' "
* Vicky Davis, 58, of Clifford Township, Susquehanna County: Pancreas, kidney.
At 37, she was diagnosed with diabetes, which through the years became progressively worse. In December 2005, she was told that her kidneys were failing. She went on dialysis in April 2006 and within a year was placed on the waiting list for a new kidney and pancreas. Initially, she says, she was told the wait would be just a few months. But whenever she received a call that there was a potential donor for her - and she says she received nine of them - the kidney and pancreas would end up going to someone else or there would be some other issue that would come up. For 3 1/2 years, she spent 3 days a week on dialysis, a process by which the blood is cleansed of toxins.
Vicky says: "Going to dialysis was like having a job. I would have to be there by 5:30 a.m. and I would not get back until 10 a.m. And it was so draining. People would ask me, 'Do you work?' And I would say, 'No, I am on dialysis.' It takes a lot out of you."
* Ramon Tejeda, 58, of Chicago: Kidney.
At 40, Tejeda had his left kidney removed because of kidney stones. In December 2003, his right kidney began to fail. It was full of cysts and functioning at only 10 percent. He began dialysis and was placed on the waiting list for a kidney. Increasingly, the 3-day-a-week, 4-hour-a-day dialysis treatments began to wear on him. Depression set in. Though he says they were keeping him alive, they were not eradicating the underlying problem he had. Unable to continue in his factory job, he went on disability. On dialysis for 6 years, he had inched to the top of the waiting list when he received word last November that Paco had died and that his kidney was being offered to him in a "directed donation." Paco was the son of his cousin, Maria.
Ramon says: "I was not doing too well. I had been on dialysis for so long. When I heard what happened to Paco, I was so very sad, very depressed. I remember him as a boy. Knowing that the kidney would come from him was hard, but it was something I knew that Paco would have wanted me to accept."
Ramon pauses and says, "He was giving me a gift."
Jesse had told Ashley that evening when he proposed: "We have been through a lot of stuff, and we will have more stuff to go through. This is not the end. But whatever happens between now and whenever, I just want you to know that I will be here for you. Whatever happens, you can count on me."
And with that he slipped the ring on her finger, which had become so bony from her weight loss that it had to be reinforced with tape to keep it from slipping off. Ashley gazed at it as her eyes pooled with tears.
Immediately, the hopelessness that had engulfed her seemed to lift. From the hallway, the nurses came into the room to admire it, one after another. Suddenly, she says she found "the courage" to go over to HUP, where she was transported later that evening. There, she and Jesse had an impromptu engagement party. He ordered in pizza and wings. What they were unaware of as they sat there eating was that Paco was slugging it out with Kennedy at the Blue Horizon, the outcome of which he had hoped would propel his boxing career into a place where he could command larger purses and better support his wife Sonia and their baby daughter, Ginette. Uncertain of when she would get the transplant she so desperately needed, Ashley said good night to Jesse and went to sleep.
Whatever else the process of organ recovery and the ensuing transplant surgeries are, it is a synchronization of many moving parts. In the case of Paco, it began when he was declared brain dead on Sunday, Nov. 22, at 7:42 p.m., at which point Janet Andrews, the transplant coordinator for the Gift of Life Donor Program, introduced herself to the Rodriguez family, offered her condolences and arranged for a priest to come by at their request. At 10:30, Andrews sat down with them and offered them the option of organ and/or tissue donation. Sonia signed the consent form an hour later. Only when that occurred could Andrews move forward. She alerted the Illinois Organ Procurement Organization of the availability of a kidney for Ramon and arranged for Paco to undergo a series of tests to evaluate his suitability to be a donor, including an echocardiogram to test his heart. Until his organs were recovered, he would remain on a ventilator with his heart beating.
On Monday at 9 a.m., GOL began the organ allocation procedure: Multiple potential recipients are identified and the organs are offered to the transplant surgeons, who assess them and reply via mobile device if they are interested or not. If they are, GOL contacts them by telephone and advises them of where they are on the list. By 1 p.m., the allocation procedure had been completed, the operating room space been reserved and the recipients had been contacted. Upbeat, Ashley says she prepared as if she was going to get better by taking a shower and braiding her hair. Told by her surgeons that they had found "a great liver," Meghan sat up in her hospital bed and said, "Let's go for it." Alexis was contacted not by her pager but by cell phone and told, "Come and get it. It's yours." At her dialysis appointment, Vicky was informed in a call and replied: "Are you sure?" Ramon could not help but think of Paco and how hard it had to be for Maria to lose a son.
That Monday at 6:30 p.m., four recovery teams entered the operating room at Hahnemann, where Paco was prepped and draped. Each organ has to be implanted within a certain span of time once it has been recovered. Says Howard Nathan, the president and CEO of GOL: "You have 3 hours for the heart, 6 for the lungs, 6 to 12 for the liver, 12 for the pancreas, and up to 48 for the kidney." In the course of the 3 1/2-hour surgery, the heart and other organs were cooled by separate cold profusion lines and are removed one by one. At 9:07, the heart was recovered, tripled bagged and transported to the adjoining operating room for Alexis. At 9:15, both lungs were recovered and rushed to HUP for Ashley. At 9:50 p.m., the liver was recovered and flown by helicopter to Johns Hopkins Hospital in Baltimore for Meghan. And at 10 p.m., both kidneys and the pancreas were recovered. A kidney and the pancreas were hurried to Geisinger Medical Center in Danville, Pa., for Vicky, and the other kidney was flown the following morning to the Rush University Medical Center in Chicago for Ramon.
Given that Paco had been a highly trained athlete, his organs were exceptional. In fact, Charlotte Owens said that the surgeon told her that he had never worked with better lungs, which Ashley discovered worked wonderfully. Suddenly, she discovered that she could breathe deeply, and that she had stopped coughing. Within weeks of their operation, the other recipients reported excellent progress. Alexis says she could hear "the profusion of blood" running through her, "that ocean sound," and that each of her senses became amplified. "I could think better," she says. "I was even answering questions off of 'Jeopardy!' " While Meghan has been hospitalized seven times since her transplant for periods ranging from 4 to 23 days and still has "dozens of tumors" in her body from her neurofibromatosis, she says she is "no longer dying but living." And Vicky and Ramon both say they have regained strength.
But curiosity set in. With the exception of Ramon, none of the others knew who the donor was. Confidentiality guidelines are such that the identities of the donor and the recipients are guarded and cannot be set aside unless either party agrees to share information. Consequently, there was always only speculation on the part of the recipients on the identity of the donor. While she was in her initial recovery, Meghan says that some friends tried to piece it together: The liver had come from Philadelphia from a 25-year-old male. When the friend told Meghan that a boxer of that age had just died in Philadelphia, she remembers thinking: "How bizarre! In this moment, I could not be fighting more."
Meghan says, "I just knew it was him. I could feel his presence."
Outside, a November rain was slanting from the gray sky in heavy sheets. But inside the third-grade classroom at Limerick Elementary School, it was dry and warm and filled with the enthusiasm of children, who were seated on the floor at the front of the room with their student-teacher, Ms. Owens. In preparation for a book the class would be beginning soon, "The One in the Middle is the Green Kangaroo," by Judy Blume, Ashley asked them to predict what certain items she placed before them would have to do with the story: A jar of peanut butter; a doll with a broken leg; a kangaroo; and a green marker. Working individually and then in groups, Ashley recorded some of the suggestions on an easel.
She stepped back to look at them and said, "These are all good predictions, but guess what? None of them are right."
"So," she continued, "we are going to have to find out what happened compared to the predictions. OK? It should be a lot of fun."
Scarcely taller than some of her students, Ashley had always hoped to become a teacher, specifically third to sixth grade. She enjoys the enthusiasm that the children bring with them to class each day. When she graduates this month from West Chester University, she plans to start looking for a teaching job in Pennsylvania or New Jersey, somewhere she and Jesse can settle down. Doctors have advised her not to teach children any younger than third grade because it would place her at an elevated risk for infections. Such warnings are heeded by her but not just because of her own health. She says she has a responsibility not just to herself but to "the gift" that she has received. To show her appreciation to Sonia, she has crocheted a pink blanket for Ginette.
It is something the others also say, that they feel a connection to Paco and his family. With the aid of Lara Moretti, the family services supervisor for GOL, Sonia reached out to the recipients in a letter to let them know who her husband was, how deeply she loved him and how she hoped that they were doing well. Says Moretti, who vets each correspondence: "Typically, families look upon the donation as a small bit of good that can come out of something terrible." One by one, the recipients replied - again, through Moretti. They told of their ordeals and of how grateful they are, how the organ they received allowed them to become fully alive and be with their loved ones. In a way that she had not anticipated, Sonia found the letters she received to be helpful to her as she had moved through the stages of grief. Sonia says she hopes that she can remain in contact with them.
"The 25 years that Francisco lived were awesome," says Sonia, as a photograph of her husband looks down at her from the dining room of his boyhood home in Chicago. "He was healthy and enjoyed life to the fullest - and now [the recipients], who have suffered for so long, have that opportunity. I want to be sure they are OK and taking care of themselves, not just because they are carrying a piece of Francisco with them but because life is supposed to be lived. I pray for them every day."
Along with what Sonia has told them, the recipients have found out more information about Paco online, where there are portions of a few of his bouts. Alexis says that she became "obsessed" with learning more about him. Given that she is a big boxing fan, she says she is surprised she had not been at the Blue Horizon for his bout with Kennedy. It was an event she would have attended, and can only think that she stayed at home that evening because of her health. But she has looked into who Paco was and says "he was no slouch," not just a fine amateur and pro boxer but a good family man, "known for joking around and laughing." Says Alexis: "I was happy to learn that he had that kind of spirit."
The bond to Paco that they feel is a deep one. In the hard days that followed her surgery, during which she experienced periods of dementia, and in her subsequent hospital stays over the course of the last year, Meghan would find herself saying, "Come on, Paco! We can do it. Work with me on this." Once, she looked down at her hands, which for a period were covered with gloves. She said to herself: "Look! I am a fighter, just like Paco!" When she had a setback in March, she rubbed the scar at the site of her incision and promised Paco: "You know, if you get me out of this, we'll go see your wife and your little girl." Meghan says she hopes to do that at some point, if only just to thank Sonia and the Rodriguez family in person.
"I feel I am not just doing it for myself now, but I am doing it for him and his family," says Meghan, who is a graduate of Elon University in North Carolina. "I want to know how Sonia is doing. I want to know how Ginette is doing. I want to go there and visit, and see the gym where Paco boxed. Me, being an athlete, I understand the dreams he had. I had wanted to be a champion. So I want to be a part of that. I would like to think of them as my extended family."
Vicky has a photo of Paco taped to her refrigerator. "When Sonia wrote me, I read her letter three times," she says. "He was so young. And she is young. But you could see there was this strong bond between them. I hope that we can become close. I would like that. Like the daughter I never had."
Vicky pauses and adds, "Somehow just saying thank you is not enough."
There will be a wedding. Bob Owens did not think he would ever have the chance to do it, but he will walk Ashley down the aisle and give her to Jesse. While there are still plans to be arranged, Ashley says she would like to have her wedding outdoors at the Valley Forge National Park and then honeymoon in Greece. Jesse says he would prefer to go to the United Kingdom, but says that Greece is fine, that he keeps telling her: "You set it up. Go where you want to go and I will follow. I want you to enjoy yourself."
Given that it is very likely she would have died were it not for the transplant, Ashley looks upon each day as precious, even if there are some worries as she moves forward. While her doctors have told her there is no physical reason she cannot have children, she is aware that the life expectancy statistics for lung-transplant patients are somewhat less encouraging than they are for the other organ recipients. "They say only 50 percent survive 5 years and 20 percent survive 10 years," says Ashley, who adds that she has also been told the absences of setbacks in the initial year are a positive indicator. But what also has her concerned is how she is going to continue to pay for the care she has to have, which includes 25 prescription drugs each day. While she and the other recipients have been covered by health plans, it has offset only a portion of the costs that they have incurred.
But Ashley does not dwell on any of this. Instead, she thinks of Paco and Sonia and Ginette; she thinks of her parents, Bob and Charlotte, and her brother, Robert; and she thinks of Jesse, who held her hand before she was wheeled into surgery and held it again when she came out. She thinks of what she can now do that she could never do before: get on a bike and go wherever she pleases; dive in a pool and hold her breath underwater; and slip on a pair of running shoes and just take off. It was something she did last March when she and her family were at Longwood Gardens. Seeing a big field stretched out before her, she challenged her brother to a race and shouted, "Daddy, take a picture!" And off she ran, the sun on her back, the wind rushing over her face.


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text:
Texas Hold'em (No Limit)
$2,500 Buy-in
Total Prize Money: $667,500
1995 Champion: Dan Harrington
1996 Champion: Mel Weiner
The Winners.
The Showdown.
1. Mel Weiner                       $250,315
Hidden Hills, CA
2. John Spadavecchia                $126,825
N Miami Beach, FL
3. Anthony Guerriero                 $63,412
Patterson, NJ
4. Russ Hamilton                     $43,388
5. Chris Ferguson                    $30,038
6. Annie Duke                        $23,363
7. Randy Holland                     $16,688
8. O'Neil Longson                    $13,350
9. Barry "Doc" Schwartz               $9,012
10. George Geary                      $9,011
11. Andre Boyer                       $8,010
12. Bill Argyros                      $8,010
Carlton, Australia
13. Marsha Waggoner                   $6,675
14. Don Kidder                        $6,675
15. Bill Henson                       $6,675
Temple City, CA
16. Jennifer Harman                   $5,340
17. Billy Thomas                      $5,340
Long Beach, MS
18. Bob Beinish                       $5,340
19. Bob Kirkeby                       $3,337
20. Scott Mayfield                    $3,337
21. Carl Mckelvey                     $3,337
22. Jim Bechtel                       $3,337
23. Dick Carson                       $3,337
24. Barbara Freer                     $3,337
25. Isai Scheinberg                   $3,337
Richmond Hill, Canada
26. Vera Blokker                      $3,337
Duivendrecht, Holland
27. John Heaney                       $3,337
The Showdown
Binion's Horseshoe, May 11, 1996 -- Mel Weiner, a 54-year-old professional poker player from Hidden Hills, California, topped a record field of 267 entrants last night to win the 21st event of the 27th annual World Series of Poker at Binion's Horseshoe Hotel and Casino. For his first-place finish in the $2,500 buy-in no-limit Texas Hold'em competition, Weiner was awarded $250,315 and a custom-designed 14-karat gold bracelet to commemorate his victory.
"It feels great to win this event," he said. "I was ecstatic to see the cards break even." This is the first World Series of Poker title for Weiner and the first time he has finished in the money in the Horseshoes annual tournament. Weiner has been playing poker since childhood.
Placing second in the $2,500 buy-in no-limit Holdem event and winning $126,825 was John Spadavecchia, a 57-year-old businessman from North Miami Beach, Florida. A regular tournament player for the past nine years, Spadavecchia holds one World Series of Poker title, as well as two titles from the Hall of Fame Poker Classic, and has finished in the money in several other major poker competitions. This marks the seventh time he has cashed in the World Series, including placing third in the 1994 championship event.
Anthony Guerriero, a 50-year-old real-estate investor from Patterson, New Jersey, won third place and $63,412. This was the first time he has entered a World Series competition.
Finishing in fourth place for $43,388 was Russ Hamilton, the 1994 World Champion of Poker. A 46-year-old professional player, Hamilton has accumulated total World Series earnings of $1,078,913 and currently ranks 12th on the tournament roster of all-time money winners.
Internet readers may be interested to know that Jeff Woods of Raleigh, NC and Jeffrey Siegal played in this tournament.
The 1996 World Series of Poker continues at the Horseshoe through May 16. The $5,000 buy-in limit Texas Holdem event begins today at noon, followed by the $1,000 buy-in Womens Seven Card Stud competition and the Press Invitational Tournament, both on May 12.
More than 4,000 entrants -- representing at least 20 countries -- are expected to participate in this years World Series, and the total money distributed is anticipated to be in excess of $11 million.
©1996, Binion's Horseshoe. Some portions ©1996, ConJelCo. All Rights Reserved.
Next Event. Previous Event
Page last modified: 5-20-96

redpajama_set_name: RedPajamaCommonCrawl


text:
/ Weekly News / Weekly News
Authored By John Nicholson February 26, 2018
Journey's Greatest Hits, which originally came out in 1989, this week became only the third album in the history of the Billboard to rack up 500 weeks on the album charts in USA. Bob Marley's Greatest Hits has chalked up 510, but do you know who has spent the most weeks on the chart? Yeah, you guessed it, Dark Side Of The Moon by Pink Floyd is way, way ahead of anyone else with an astonishing 937 weeks!! That means Journey will need to be on the charts for over eight more years to get close to them.
For such a long charting record, Journey's Greatest Hits peaked at a modest #10. It's been one of those albums that just sells and sells but has never had a really stellar sales week to push it to #1. Mind you, I don't suppose the band are complaining seeing as it's already sold 15 million in USA alone.


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Bara Mamadou Lamine Ndiaye oder kurz Bebeto (* 31. Dezember 1991 in Tivaouane) ist ein senegalesischer Fußballspieler.

Karriere
Ndiaye wechselte 2008 zu Teramo Calcio und spielte hier ein Jahr für die Nachwuchsabteilung. Anschließend begann er für die Profimannschaft aufzulaufen. In der Saison 2010/11 spielte er dann bei Giulianova Calcio und wechselte zur Saison 2011/12 zum FC Lugano. Bei diesem Verein blieb er die nächsten drei Spielzeiten unter Vertrag und wurde zwischenzeitlich an die ungarischen Vereine Kaposvári Rákóczi FC und Kecskeméti TE ausgeliehen. In der Sommertransferperiode 2014 wechselte er zu Letzterem.

Zur Saison 2015/16 wechselte Ndiaye die türkische TFF 1. Lig zu Gaziantep Büyükşehir Belediyespor. Nach dieser Spielzeit verließ er diesen Verein wieder und zog zum zypriotischen Verein Doxa Katokopia weiter.

Weblinks

Fußballspieler (Doxa Katokopia)
Fußballspieler (FC Lugano)
Fußballspieler (Gaziantep FK)
Fußballspieler (Giulianova Calcio)
Fußballspieler (Kaposvári Rákóczi FC)
Fußballspieler (Kecskeméti TE)
Fußballspieler (Teramo Calcio)
Senegalese
Geboren 1991
Mann

redpajama_set_name: RedPajamaWikipedia


text:
Soup Moat is bigger, better, sludgier than ever on 'Be Brave, Run Away'
What's new, and what's coming soon to the East Side
331 Friday fish frys and counting: Don's Diner & Cocktails / Grocery & Liquor curbside pickup (Milwaukee)
Ron Funches talks stand-up, acting, voiceover, new show with Kesha before Milwaukee tour stop
Take a behind-the-scenes peek at the ongoing Oriental Theatre restoration
Explore the all-too-timely world of conspiracy theories with Tea Krulos' new book, 'American Madness'
'BUCKS IN SIX' captures revelry, randomness, river diving following Bucks' big win
My First Band: Adam Fletcher (The Copyrights)
Here's our Green Bay Packers All Holiday Team
In case you were somehow unaware, Christmas is right around the corner (as well as other wintertime holidays, which are either happening now or will be going on soon). That's right, in just a couple days, the stockings will be stuffed, the presents will be piled up and ready to be ripped open, tables will be covered by crock pots and snacks, and family members will be doing their damnedest to avoid bringing up any aspect of the increasingly nightmarish world in which we live.
Thankfully, this Christmas offers the entertainment option and the political/pandemic deflecting distraction of a Packers game…at least if nobody brings up Aaron Rodgers interviews or the NFL's abysmal handling of the COVID crisis. Yes, Green Bay will host what's left of the Cleveland Browns at Lambeau Field this Saturday for a Christmas afternoon contest. Since we're in the seasonal spirit this year, we decided to comb through the franchise's rosters in order to put together a Green Bay Packers All Holiday Team composed of both past and present players. Feel free to add anyone you think we missed in the comments.
Happy holidays and Go Pack Go!
Ed Bell
Nothing says Christmastime and the winter holiday season in general quite like the presence of bells. There's Jingle Bells, Silver Bells, Carol Of The Bells, "every time a bell rings, an angel gets its wings" and so on. In the realm of Packers football, there was also an offensive lineman named Ed Bell who played with Green Bay for three seasons in the late 1940s.
Ken Bowman
When it comes time to wrap presents, it's important to have a reliable Bow-man. Among Packers players with the word "Bow" in their last name, two-time Super Bowl champion center Ken Bowman was the best of the bunch. Early aughts safety Matt Bowen and short-tenured defensive end David Bowens are the other players in this festive, Bow-based group.
Brandon Chillar
"Chillar" sort of sounds like Chiller. The holiday season is chilly. There's a player named Brandon Chillar who was a linebacker on the 2010 Super Bowl-winning team. It kind of works…what more do you people want from me?!
Santana Dotson
Both Santa Claus and SANTAna Dotson are famous for their sacks. The former fills his sack with toys to distribute to children all around the world, whereas the latter tallied a total of 26 quarterback sacks during his six seasons with the Packers.
Derrick Frost
Derrick Frost averaged 42.1 yards per punt during the 2008 season, his one and only year with Green Bay. More importantly, at least in terms of this Packers article about winter and the holidays, his last name is Frost.
Vonnie Holliday
Even though his last name isn't the correct spelling, Vonnie Holliday gave Packers fans a lot of reasons to celebrate during his five seasons in Green Bay. During that span, he amassed 32 sacks and a pair of interceptions.
Carlyle Holiday
Packers reserve wide receiver and emergency quarterback Carlyle Holiday didn't have nearly the career that Vonnie Holliday did, but his last name is spelled right, so he has that going for him.
Johnny Jolly
With all due respect to early-'80s defensive back Mike Jolly (whom I just learned existed when I started putting this article together), there's no Packers player more Jolly and more worthy of inclusion on this list than defensive lineman Johnny Jolly.
Much like the Wise Men followed the North Star to Bethlehem in the bible or whatever, the Packers teams of the 1950s and '60s followed quarterback Bart Starr to victory. Take a moment to look at his career stats. Do you see what I see? A Starr, a Starr in the Hall Of Fame. Green Bay fans will remember his name.
Blaise Winter
Remember former Packers defensive lineman-turned-motivational speaker Blaise Winter? "I think so," you say? Well, he exists and, yes, his last name is Winter—the very same season in which Christmas occurs!
Frank Winters
Between late December and mid-March, weather conditions in Green Bay and the rest of Wisconsin can be cold, unforgiving, and quite direct. I guess you could say the state is known for its Frank Winters.
Tom Brown and Jerry Kramer
Safety Tom Brown and Hall Of Fame guard Jerry Kramer were both important parts of some teams in Lombardi era Packers dynasty. Similarly, the Tom & Jerry cocktail is an important part of the Wisconsin holiday tradition.
Kevin King and Kingsley Keke
Current Packers teammates Kevin King and Kingsley Keke are just one "King"-named player short of being the Three Kings Of Titletown.
Charlie Hall, Korey Hall, Lamont Hall, and Mark Hall
Deck the Halls with former Packers. Fa la la la la, la la la la.
Extra PointsFeatureGreen Bay Packers
Before co-founding Milwaukee Record, Tyler Maas wrote for virtually every Milwaukee publication (except Wassup! Magazine). He lives in Bay View and enjoys both stuff and things.


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SchoolKart is a one-stop shop for all the school needs. The website carries uniforms, books, and school essentials for more than 500 schools. With cash on delivery option, SchoolKart delivers across India. The portal is user-friendly. Deals are divided into price slabs so you can shop according to your budget. You can also place bulk orders on SchoolKart. The website specialises in the delivery of uniform and books of all NCR schools.
Very useful information. Itsy bitsy is very nice place to shop craft and project materials for school kids. Go through the others too.


text:
Q: Making a working neuronal network I've got the following two MWE:
\documentclass[border=5pt,tikz]{standalone}
   \begin{document}

        \begin{tikzpicture}

        %%% %%% %%%
        %>> We are building a perceptron
        %%% %%% %%%

            % input vectors
            \xdef\firstinputarray{{1,2}}
            \xdef\secondinputarray{{4,8}}

            % output vector
            \xdef\outputarray{{1,0}} % The 1 means that for \firstinputarray the statement is true, for \secondinputarray is wrong

            % weights
            \xdef\zerow{.5}
            \xdef\firstw{2}
            \xdef\secondw{1.75}

            % bias
            \xdef\bias{1}

            % Heaviside

            \newcommand{\heaviside}[1]{
                \ifnum#1>0
                    \pgfmathsetmacro{\a}{1}
                \fi
                \ifnum#1=0
                    \pgfmathsetmacro{\a}{1}
                \fi
                \ifnum#1<0
                    \pgfmathsetmacro{\a}{0}
                \fi
            }

            % Floor function
            \newcommand{\floor}[1]{
                \pgfmathsetmacro{\floornumber}{int(floor(#1))}
            }

            \newcommand{\forheavi}[1]{
                \pgfmathsetmacro{\forheavinumber}{int(floor(#1))}
            }

            % new weights
            \pgfmathsetmacro{\firstway}{\zerow*\bias+\firstinputarray[0]*\firstw+\firstinputarray[1]*\secondw}

            \forheavi{\firstway}
            \heaviside{\forheavinumber}

            \node (a) {\texttt{firstway}: \pgfmathprintnumber{\firstway}\qquad$\mathrm{H}(\mathtt{firstway})$:\pgfmathprintnumber{\a}};

            % set new weights
            \ifnum\a>0
                \pgfmathsetmacro{\firstw}{.5*(\firstway-0)*\firstinputarray[0]}
            \fi
                \forheavi{\firstway}
                \heaviside{\forheavinumber}
            \node[above left,yshift=1cm] at (a.north east) {new \texttt{firstw}: \pgfmathprintnumber{\firstw}\qquad$\mathrm{H}(\mathtt{new fistw})$: \pgfmathprintnumber{\a}};

    %           \forheavi{0}
    %           \heaviside{\forheavinumber}
    %               \node[red,draw,below=.5cm] at (a) {\pgfmathprintnumber{\a}};

        \end{tikzpicture}

    \end{document}

and

    \documentclass[border=5pt,tikz]{standalone}
    \usetikzlibrary{decorations.pathreplacing}
    \tikzset{
        neuron/.style={
            draw,circle,inner sep=.3cm,fill=white
        },
        brace/.style={
            decorate,decoration={brace,amplitude=.3cm},thick
        }
    }
    \begin{document}
        \begin{tikzpicture}
            \foreach \x in {0,1,...,8}
            {
                \node[neuron] (a\x) at (0,\x) {};
            }
            \foreach \y in {-4,-3,...,12}
            {
                \node[neuron] (b\y) at (8,\y) {};
            }
            \foreach \z in {-2,-1,...,10}
            {
                \node[neuron] (c\z) at (16,\z) {};
                \pgfmathsetmacro{\n}{10-\z}
                    \draw[->] (c\z) --+ (1,0) node[right] {\pgfmathprintnumber{\n}};
            }
            \foreach \x in {0,1,...,8}
            \foreach \y in {-4,-3,...,12}
            \foreach \z in {-2,-1,...,10}
            {
                \draw[->] (a\x) -- (b\y);
                \draw[->] (b\y) -- (c\z);
            }
            \draw[brace] ([xshift=-1cm]a0.south) -- ([xshift=-1cm]a8.north) node[midway,text width=2.5cm,left=.5cm] {
                \begin{tabular}{c}
                    input layer \\
                    (784 neurons)
                \end{tabular}
            };
            \node[above=1cm] at (b12) {
                \begin{tabular}{c}
                    hidden layer \\
                    ($n = 15$ neurons)
                \end{tabular}
            };
            \node[above=1cm] at (c10) {
                \begin{tabular}{c}
                    output layer
                \end{tabular}
            };
        \end{tikzpicture}
    \end{document}

with the following output:

I've got the following questions:

*

*How can I achieve that I can enter the input arrays and the starting weights and the first code does the calculations (more precisely: how can I "extend" the foreach loop so that the algorithm stops when the right resulst is achieved) and

*How can I than highlight the weights visual e.g. I let TikZ draw two nodes (which represent neurons) and the more "important" a bond is the line (which connects the two neurons) gets thicker (this all happens step for step, so that we can create a little animation)?

P.S.: The feedforward network doesn't need to have such a big amount of neurons, this is just an illustration.

redpajama_set_name: RedPajamaStackExchange


Desert Mountain Energy Corp. (TSX.V: DME) is engaged in exploration and development of helium, oil & gas and mineral properties, in the Southwestern United States. At PDAC2019, we learned, from Irwin Olian, President and CEO of Desert Mountain Energy, the Company's current focus is on helium, which is in surging demand in the high-tech world. There is currently a worldwide shortage of helium, its price has gone up 135%, in the last 12 months at public auction, to the equivalent of $279 per thousand cubic feet of crude helium gas. Desert Mountain's flagship Heliopolis Project comprises 39,742 acres of key helium prospects, located in prolific Holbrook Basin, in East Central Arizona, widely considered to be the world's best address for helium. Near term plans include a drilling program, late in the second quarter this year.
Seabridge Gold (TSX: SEA, NYSE:SA) is a gold exploration company, designed to provide its shareholders with exceptional leverage to a rising gold price. The Company holds one of the world's largest resource bases of gold, copper and silver, in keeping with its prime objective of growing resource and reserve ownership per share. We learned from Rudi Fronk, who is Co-Founder, Chairman, and CEO of Seabridge Gold, that they are sitting with 45 million ounces of proven or probable gold reserves, most of which are situated under their KSM asset, located in northern British Columbia. KSM is one of the largest undeveloped gold-copper projects on the planet. With copper and silver credits, the all-in sustaining costs at KSM will be about $350, which would be one of the industry's lowest all-in cost.
Lepidico Ltd (ASX: LPD) is a lithium exploration and development company, who's strategic objective is to become a fully integrated lithium business from mine to battery-grade lithium chemical. At PDAC2019, we learned from Joe Walsh, Managing Director of Lepidico, that Lepidico is a unique investment opportunity in the lithium space, as the 100% owner and developer of the L-Max® and LOH-MaxTM technologies, proprietary processes, which have the potential to produce commercially, either lithium hydroxide or lithium carbonate, from unconventional sources. These technologies have the potential to place Lepidico at the forefront of the lithium industry.
Paramount Gold Nevada Corp. (NYSE American: PZG) owns 100% of the Grassy Mountain Gold Project, which consists of approximately 11,000 acres, located on private and BLM land in Malheur County, Oregon. The project contains a gold-silver deposit (100% located on private land) for which results of a positive PFS have been released in 2018 and key permitting milestones accomplished. At PDAC2019, we learned from John Seaberg, Executive Chairman of Paramount Gold Nevada, that recently they acquired the Frost Project 12 miles away from the Grassy Mountain. Plans for 2019 include completing the permitting process for the Grassy Mountain, which is going to be a small, but very profitable operation and will have tremendous economic impact on the local communities of Vale and Ontario that are very supportive of the project.
White Gold Corp. (TSX.V: WGO, OTC – Nasdaq Intl: WHGOF, FRA: 29W) owns a portfolio of 22,040 quartz claims across 35 properties, covering over 439,000 hectares, representing over 40% of the Yukon's White Gold District. White Gold is one of the most active exploration companies in Canada, which last year led to 4 new gold discoveries, including the Vertigo, which intersected 22.47 g/t gold over 30.46m from surface. Their flagship White Gold property has a mineral resource of 960,970 oz Indicated at 2.43 g/t gold and 282,490 oz Inferred at 1.70 g/t gold. At PDAC 2019, we learned from White Gold's David D'Onofrio, CEO, and Shawn Ryan, Chief Technical Advisor, that in 2019, they are focused on diamond drilling to expand their Vertigo discovery, grow their Golden Saddle deposit, increase their recently acquired 230,000 oz VG resource, as well as identify and test multiple high priority regional targets.
Revival Gold Inc. (TSXV: RVG, OTCQB: RVLGF) is a growth-focused gold exploration and development company. The Company has the right to acquire a 100% interest in Meridian Beartrack Co., owner of the former producing Beartrack Gold Project, located in Lemhi County, Idaho. Revival also owns the rights to a 100% interest in the neighboring Arnett Gold Project. At the 2019 Vancouver Resource Investment Conference, we learned from Hugh Agro, President and CEO of Revival Gold, that in 2018 they did about 8,600 meters of drilling, and the Company's geologists are working on new drilling targets. In 2019, Mr. Agro plans to continue exploration and metallurgical test work, and by the end of the year initiating an updated resource estimate for the project. According to Mr. Agro, investors should focus on finding a winning and committed management team, with the specific experience to be able to take the project forward.
Fission Uranium Corp. (TSX: FCU, OTCQX: FCUUFÂ, FRANKFURT: 2FU) is focused on exploration and development of the Patterson Lake South (PLS) uranium property, located in Canada's Athabasca Basin, home to the world's richest uranium mines. The project is host to the Triple R deposit - the most significant high-grade shallow depth deposit in the region. At PDAC2019, we learned from Ross McElroy, who is President and COO, and Dev Randhawa, who is CEO and Chairman of Fission Uranium, that while 2018 saw the uranium supply going down, Fission has been working on their PFS that is expected to be ready in early Q2 2019, after which the company will go on to a bankable feasibility study. According to Mr. Randhawa, Fission Uranium is a well-funded company without debt, that boasts a highly decorated management team. There's a uranium deficit, and eventually as surplus gets eaten up, uranium prices have to go up.
Ascot Resources Ltd. (TSX.V: AOT, OTCQX:AOTVF) is a gold and silver focused exploration company, with a portfolio of advanced and grassroots projects in the Golden Triangle region of British Columbia. The company's flagship Premier project is a near-term, high-grade advanced exploration project, with large upside potential. We learned from Derek White, President and CEO of Ascot, that in early January they announced a deal to acquire IDM Mining, with its feasibility-ready, shovel-ready high-grade Red Mountain project. According to Mr. White, the deal places Ascot in the top 7% of the world in gold grade, which is critical for lowering operation costs. According to Mr. White, the main reason investors should look forward to Ascot/IDM merger is the critical mass - a pathway to 200,000-plus a year for 10 years of high-grade low-cost production.
Integra Resources Corp. (TSXV:ITR ; OTCQX:IRRZF) is a development-stage company, focused on advancing its DeLamar Project, consisting of the neighbouring DeLamar and Florida Mountain Gold and Silver Deposits, in the heart of the historic Owyhee County mining district, in south western Idaho. The first exploration program, in over 25 years, began on the DeLamar Project in 2018, with more than 23,000 meters drilled. The management team comprises the former executive team from Integra Gold Corp. At the PDAC2019 conference, we learned from Chris Gordon, Business Development with Integra Resources, that this year they expect to grow the resource, move a good part of the inferred to the indicated category, analyze the metallurgical studies, and, by September, do a PEA. According to Mr. Gordon, Integra is being very diligent to maintain a tight share structure and an excellent list of shareholders.
Avalon Advanced Materials Inc. (TSX: AVL, OTCQB: AVLNF) is currently focusing on developing its Separation Rapids Lithium Project, Kenora, ON and its East Kemptville Tin-Indium Project, Yarmouth, NS to production, while continuing to advance its Nechalacho Rare Earths asset. At PDAC2019, we learned from Don Bubar, President and CEO of Avalon Advanced Materials, that they were an early mover in the space of the advanced materials and have created a portfolio of assets, with a broad range of exposure to the emerging new clean tech materials. Avalon's overall strategy is to be in a position to bring a new supply to the market, as new technology creates new demand. According to Mr. Bubar, it has taken a while for the circumstances to be perfectly aligned, and right now Avalon finds itself in the great position of having three projects coming together at the same time.
Nevada Exploration Inc. (TSX-V: NGE; OTCQB: NVDEF) is integrating hydrogeochemistry, with conventional exploration tools, to develop a Nevada-specific regional-scale geochemistry exploration program. Using its proprietary technology, Nevada Exploration has completed the world's largest groundwater sampling program for gold exploration, to evaluate Nevada's covered basins for new gold exploration targets. The Company then developed its Scorpion drill rig, a small-footprint, truck-mounted, small-diameter RC drill rig, specifically tailored to the drilling conditions in Nevada's basins. We learned from James Buskard, President of Nevada Exploration, that in 2019 they're looking to advance all three of their major projects: South Grass Valley, Grass Valley, and Kelly Creek.
At the PDAC2019, we learned from Heye Daun, President and CEO of Osino Resources Corp. (TSXV: OSI), a gold exploration company, active in Namibia, that they are currently focused on their drill-ready Twin Hills project, which is a new discovery of a very large gold system right next to an established producing gold mine. Plans for 2019 include financing and exploration drilling at Twin Hills, with the goal of turning the discovery of a big gold system into a real economic discovery. Osino's secondary project is an advanced stage Goldkuppe exploration project and a group of target generation stage licenses that are contiguous to B2Gold's Otjikoto Gold Mine. According to Mr. Daun, Osino is in discussions with possible JV partners to develop its non-core projects.
Aethon Minerals Corp. (TSX-V: AET) is a well-funded mineral exploration company, focused in South America. They were spun out of Altius Minerals, in May of 2018, with over 100,000 hectares of prospective land package, in northern Chile and a goal to advance it and to find potential partners to explore the properties. We learned from John Miniotis, Interim CEO of Aethon Minerals, that the Company recently entered into an option agreement to acquire an interest in the advanced-stage Diablillos silver-gold exploration project, owned by AbraPlata Resources in Argentina. The project has over 80 million ounces of silver and 750,000 ounces of gold, in measured and indicated resources. Near term plans include doing detailed due diligence on the Diablillos property and working out an option agreement with favorable terms.
Abitibi Royalties Inc. (TSX-V: RZZ) owns various royalties at the Canadian Malartic Mine, near Val-d'Or Québec, and is building a portfolio of royalties on early stage properties, near producing mines. At PDAC 2019, we learned from Ian Ball, President and CEO of Abitibi Royalties that their royalty income started at the end of last year and the Company is looking at $4 to $4 1/2 million in cashflow this year. The Company is in a strong financial position and is debt free, and they bought back about $1 to $1 1/2 million worth of shares over the past 12 months.
Largo Resources Ltd. (TSX: LGO, OTCQX: LGORF) is a Toronto-based strategic mineral company, focused on the production of vanadium flake, high purity vanadium flake and high purity vanadium powder at the Maracás Menchen Mine located in Bahia State, Brazil. At PDAC2019, we learned from Mark Smith, CEO and Director of Largo Resources, that Largo's team is one of the best in the industry, they are committed and competent, and they are developing the world's best vanadium resource, processing it with the lowest unit cost, highest recovery, and ending up with the highest purity level of V205 in the world. According to Mr. Smith, today's fundamental supply deficit in the vanadium markets is very favorable to the company, which recorded cash flow of about $129 million dollars in the third quarter of 2018.
Novo Resources Corp. (TSX-V: NVO; OTCQX: NSRPF), a Canadian gold exploration and development company, has built up a significant land package covering approximately 12,000 sq km with varying ownership interests in the Pilbara region of Western Australia. We learned from Dr. Quinton Hennigh, President and Chairman of NOVO Resources Corp., that they are looking for world-class large-scale conglomerate gold deposits, and their most advanced project is called Beatons Creek, near the town of Nullagine, in the East Pilbara region. At the moment, Novo is focused on increasing significantly, the current resource of about 670,000 ounces. Novo is working on another interesting project, the brand new conglomerate gold Karratha discovery, in the West Pilbara region, very large system, where the Company did aggressive bulk sampling in 2018. The Company is tightly held with such major shareholders as Kirkland Lake Gold, Newmont Mining, Eric Sprott, and Mark Creasy.
West Red Lake Gold Mines Inc. (CSE: RLG, OTCQB: RLGMF, FSE: HYK) has assembled a 3100-hectare property, which has a 12 kilometer strike length and 3 former gold mines, in the prolific Red Lake Gold District of Northwestern Ontario, Canada. At PDAC 2019, we learned from John Kontak, who is the President of West Red Lake Gold Mines, that, in 2018, they drilled below the 500 meters of the resource area at the high-grade Rowan Mine Deposit, a joint venture with Goldcorp, with the future goal to increase its current 1.087 million ounces of inferred resource. Another drilling program was conducted at the high-grade NT Zone, with good results for what could be a whole new deposit. Plans for 2019 include more exploration at the Rowan Mine Deposit and the NT Zone, with the goal to develop it into a high grade gold deposit. The company has an experienced Management team that has explored and developed and successfully sold gold assets in Ontario in the past.


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There check for the Important Notification section which will be available.
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text:
\section{Introduction}
Let $f(x)$ be the common probability density function of a sequence $\{X_n\}_{n=1}^{\infty}$ of identically distributed observations. The associated Shannon entropy
\begin{align} \label{sf}
S(f) = {{\mathbb E}\,} [-\log f(X) ] = -\int f(x)\;\log f(x) \; dx
\end{align}
of such an observation was first introduced by Claude Shannon \cite{Shannon}.  In his 1948 paper, Shannon utilized this tool in his mathematical investigation of the theory of communication.  Today, entropy is widely applied in the fields of information theory, statistical classification, pattern recognition and so on, since it is a measure of the amount of uncertainty present in a probability distribution.

In the literature, several estimators for the Shannon entropy have been introduced.  See Beirlant et al. \cite{Beirlant} for an overview.  Many of these estimators have been studied in cases where the data is independent.  In 1976, Ahmad and Lin \cite{Ahmad} obtained results using the resubstitution estimator $H_n = -\frac{1}{n} \sum_{i=1}^{n} \ln f_n (X_i)$ for independent data $\{X_i\}_{i=1}^n$.  In particular, he showed consistency in the first and second mean under certain regularity conditions.  Here, $f_n(x)$ is the kernel density estimator.  Dmitriev and Tarasenko \cite{Dmitriev} reported results in 1973 for estimating functionals of the type $\int H\big(f(x), f'(x), ..., f^{k}(x)\big) dx$, where the common density $f(x)$ of the independent $X_i$ is assumed to have at least $k$ derivatives.  Plugging in kernel density estimators (see their paper and references therein) for the arguments of $H$ and integrating only over the symmetric interval $[-k_n, k_n]$, which is determined by a sequence $\{k_n\}_{n=1}^\infty$ of a certain order, they provided a result for the estimation of Shannon entropy using the estimator that Beirlant et al. \cite{Beirlant} refer to as the integral estimator.  Their results give conditions for almost sure convergence.

Interestingly enough, Dmitriev and Tarasenko \cite{Dmitriev} also provided (because their work is a more general investigation of functionals) a result for the estimation of the quadratic R\'{e}nyi entropy $Q(f)=\int f^2(x) dx$.  Conditions are provided specifically for the almost sure convergence of their estimator to the true value $Q(f)$.  The estimation of R\'{e}nyi entropy for the dependent case is challenging.  A dependent case is treated by Sang, Sang, and Xu \cite{Sang}.  They studied the estimation of the quadratic Entropy for the one-sided linear process.  Utilizing the Fourier transform along with the projection method, they demonstrate that the kernel entropy estimator satisfies a central limit theorem for short memory linear processes.

To study the Shannon entropy for dependent data is also a challenging problem, and to the best of our knowledge, general results for the Shannon entropy estimation of regular time series data are still unknown.  In this paper, we study the Shannon entropy $S(f)$ for the one-sided linear process
\begin{align}\label{lp}
X_n=\sum\limits_{i=0}^\infty a_i \varepsilon_{n-i},
\end{align}
where the innovations $\varepsilon_i$ are independent and identically distributed real valued random variables on some probability space $(\Omega, \mathcal{F}, {\mathbb P})$ with mean zero and finite variance $\sigma_\varepsilon^2$ and where the collection $\{a_i : i\ge 0 \}$ of real coefficients satisfies $\sum\limits_{i=0}^\infty a_i^2 < \infty$.  Additionally, we will require that the common density $f_{\varepsilon} (x)$ of the innovations be bounded.  The estimator we utilize employs the kernel method, which was first introduced by Rosenblatt \cite{R} and Parzen \cite{Parzen}.  The kernel estimator will be denoted by
\begin{align} \label{kernel_est}
f_n(x) = \frac{1}{n h_n} \sum_{i=1}^{n} K \left( \frac{x - X_i}{h_n} \right),
\end{align}
where the sequence $ \{h_n\}_{n=1}^{\infty} $ provides the bandwidths, and $ K: {\mathbb R} \to {\mathbb R} $ is the kernel function which satisfies $\int_{\mathbb R} K(x)dx=1$.  Typically, the kernel function is a probability density function.

This method has proven to be successful in estimating probability density functions and their derivatives, regression functions, etc. in both the independent and dependent setting.  For the independent setting, see the books (Devroye and Gy\"{o}rfi \cite{DG}; Silverman \cite{Silverman}; Nadaraya \cite{Nadaraya}; Wand and Jones \cite{WJ};  Schimek \cite{Schimek}; Scott \cite{Scott}) and the references therein.  For the dependent setting, we refer the reader to (Tran \cite{Tran}; Honda \cite{Honda}; Wu and Mielniczuk \cite{WM}; Wu, Huang and Huang \cite{WHH}). Bandwidth selection is an important issue in kernel density estimation, and there is much research in this direction. See, e.g., Duin \cite{D}, Rudemo \cite{R} and Slaoui \cite{S1, S2}.

A few remarks about notation and terms used in the paper follow.  Let $\{a_{n}\}_{n=1}^{\infty}$ and $\{b_{n}\}_{n=1}^{\infty}$ be real-valued sequences.
 By $a_{n}=o(b_{n})$ we understand that $a_{n}/b_{n}\rightarrow0$ and $a_{n}=O(b_{n})$ means that $\lim\sup|a_{n}/b_{n}|<C$ for some positive number $C$. Essentially, this is the standard Landau little oh and big oh notation.  When we write, $a_n \ll b_n$, we mean $a_n = o(b_n)$, and as one might guess, $b_n \gg a_n$ means $a_n \ll b_n$.  Also, we employ the notation $a_n \asymp b_n$ to indicate that $0 < \liminf_{n\to\infty} \frac{a_n}{b_n} \leq \limsup_{n\to\infty} \frac{a_n}{b_n} < \infty$.
A function $l:[0,\infty)\rightarrow{\mathbb{R}}$ is referred to as \textit{slowly varying} (at $\infty$) if it is positive and measurable on $[A,\infty)$ for some $A\in{\mathbb{R}}^{+}$ such that $\lim\limits_{x\rightarrow\infty}l(\lambda x)/l(x)=1$ holds for each $\lambda\in{\mathbb{R}}^{+}$.  The set of all functions $g:{\mathbb R}\to{\mathbb R}$ which are \textit{H\"{o}lder continuous} of some order $r$ will be denoted as $\mathcal{C}^r({\mathbb R})$.  That is, for each $g\in\mathcal{C}^r({\mathbb R})$ there exists $C_g\in{\mathbb R}^+$ such that for all $x, x' \in {\mathbb R}$, we have $|g(x)-g(x')| \leq C_g |x-x'|^r$, and when $r=1$, we recognize this as the well-known \textit{Lipschitz condition}.  The notation ${\mathcal{L}}^p(E)$ with $0<p<\infty$ represents the set of all real-valued functions $f$ defined on some measure space $(E,\mathcal{A},\mu)$ having the property that $\int_E |f(x)|^p\,d\mu<\infty$.  In the case that $E={\mathbb R}$ and unless otherwise specified, the measure $\mu$ is tacitly understood to be Lebesgue measure and $\mathcal{A}$ is assumed to contain the Borel sets.  ${\mathcal{L}}^\infty(E)$ refers to the set of real-valued functions defined on $E$ which are bounded almost everywhere.  Whenever the domain space of the function is understood, we may simply write ${\mathcal{L}}^p$.

\newpage
The following are \textit{bandwidth, kernel, and density conditions} that we shall refer to throughout this paper.
\begin{description}
	\item [\textbf{B.1}] $ h_n \asymp (n^{-1} \log n)^\frac{1}{5}$.

	\item [\textbf{K.1}] $K\in\mathcal{C}^\iota({\mathbb R})$ for some $\iota\in(0,1]$ is bounded with bounded support.
	\item [\textbf{K.2}] $ \int u K(u) \ du =0 $.
	\item [\textbf{D.1}] $ f_\varepsilon, f'_\varepsilon, f''_\varepsilon \in {\mathcal{L}}^\infty({\mathbb R}) $.
	\item [\textbf{D.2}] $ f_\varepsilon, f'_\varepsilon, f''_\varepsilon \in {\mathcal{L}}^2({\mathbb R}).$
	\item [\textbf{D.3}] $ f'' \in {\mathcal{L}}^\infty({\mathbb R}) $.
\end{description}
Notice that the bandwidth, kernel, and density conditions are prefixed using \textbf{B}, \textbf{K}, and \textbf{D}, respectively.

In this first section, we have provided an introduction to the problem, a survey of past research in this area, and the notation to be used throughout.  The main results are reported in section two.  In section three, we present the proofs of the main results. Finally,  the appendix introduces the reader to foundational results which will be required in the proof of our main results.

\section{Main Results}\label{main}
If $\{\varepsilon_{i}:\; i\in{\mathbb Z}\}$ is a sequence of independent and identically distributed random variables over a common probability space $(\Omega, \mathcal{F}, {\mathbb P})$ in ${\mathcal{L}}^{q}(\Omega)$ for some $q>0$, ${{\mathbb E}\,}\varepsilon_{i}=0$ when $q\geq 1$, and $\{a_{i}\}^{\infty}_{i=0}$ is a sequence of real coefficients such that $\sum\limits_{i=0}^{\infty}|a_{i}
|^{2\wedge q}<\infty$, then the linear process $X_n$ given in (\ref{lp}) exists and is well-defined. For the case $q\geq 2$ where the innovations have finite variance, we say that the process has \textbf{short memory (short range dependence)} if $\sum\limits_{i=0}^{\infty}|a_i|<\infty$ and $\sum\limits_{i=0}^{\infty} a_i\neq 0$ and \textbf{long memory (long range dependence)} otherwise.  Throughout, we assume that each $\varepsilon_{i}\in {\mathcal{L}}^q$ with $q\geq 2$

Let $f(x)$ be the probability density function of the linear process $X_n=\sum\limits_{i=0}^\infty a_i\varepsilon_{n-i}$, $ n \in \mathbb{N}$ defined in (\ref{lp})
\;\;In this paper, we estimate the Shannon Entropy $-\int f(x) \log f(x)\;dx$ of the linear process
\;\;To do this, we employ the integral estimator
\begin{align} \label{IntEst}
S_n(f) = - \int_{A_n} f_n(x)\;\log f_n(x)\;dx,
\end{align}
where $ f _n(x) $ is the standard kernel density estimator defined in (\ref{kernel_est}).  The (random) sets $A_n$ are given by
\begin{align} \label{A_n}
A_n = \{ x \in {\mathbb R} : 0 < \gamma_n \leq f_n (x) \},
\end{align}
where $\{\gamma_n\}_{n=1}^{\infty}$ is an appropriately defined sequence in ${\mathbb R}^+$ that converges to zero.

Our estimator utilizes the kernel method of density estimation, and we will accordingly require adherence of the kernel to certain conditions.  In addition, we impose some conditions on the bandwidths and on some of the densities of the problem.  These conditions were listed in the previous section.  Based on these conditions, let us consider the properties of the estimator (\ref{IntEst}).  We proceed in a manner similar to the analysis done by Bouzebda and Elhattab \cite{BE} for the independent case.

\begin{theorem} \label{thm1}
	Let $\{X_n : n\in{\mathbb N}\}$ be the linear process given in (\ref{lp}), and assume that it has short memory.  Furthermore, assume that $S(f)$ is finite.

	If the bandwidth, kernel, and density conditions listed earlier are satisfied, then
	\begin{align*}
	\limsup_{n\to\infty}\;\bigg(\frac{n\;\gamma_n^5}{\log n}\bigg)^\frac{2}{5}\;\;\bigg|\;S_n(f) - \int_{A_n} \big(-\log f(x)\big) f(x)\;dx\;\bigg|
	\end{align*}
	is bounded almost surely whenever the condition $ \gamma_n \gg h_n $ is imposed on the sequence $\{\gamma_n\}_{n=1}^{\infty}$.
\end{theorem}
\begin{corollary} \label{cor1}
	If the conditions of Theorem \ref{thm1} hold, then we have
	\begin{align*}
	\lim_{n\to\infty} |S_n(f) - S(f)| = 0
	\end{align*}
	almost surely.
\end{corollary}

\begin{theorem} \label{thm2}
	Let $\{X_n : n\in{\mathbb N}\}$ be the linear process given in (\ref{lp}), and assume that it has short memory.  Furthermore, assume that $S(f)$ is finite.

	If the bandwidth, kernel, and density conditions listed earlier are satisfied, then
	\begin{align}
	\limsup_{n\to\infty}\;\bigg(\frac{n\;\gamma_n^5}{\log n}\bigg)^\frac{2}{5}\;\;\bigg\lVert\;S_n(f) - \int_{A_n} \big(-\log f(x)\big) f(x)\;dx\;\bigg\rVert_2
	\end{align}
	is bounded whenever the condition $ \gamma_n \gg h_n $ is imposed on the sequence $\{\gamma_n\}_{n=1}^{\infty}$.
\end{theorem}

\begin{corollary} \label{cor2}
	If the conditions of Theorem \ref{thm2} hold, then the mean squared error (MSE) satisfies
	\begin{align*}
	\lim_{n\to\infty} \textup{MSE} (S_n(f)) = 0.
	\end{align*}
\end{corollary}

\begin{remark}
In this paper we work on the entropy estimation for short memory linear processes by applying the integral method. It is interesting to know whether the similar results hold for long memory linear processes. It is also interesting to know whether the resubstitution method works for dependent data such as linear processes. However, the research in these directions are beyond the scope of this paper. We leave the research in these directions for future work.
\end{remark}

\begin{remark}
In a wide range of disciplines, including finance, geology, and engineering, many time series may be modeled using a linear process. In such instances, our result provides a method for estimating the associated Shannon Entropy.  One example is the discriminatory data on the arrival phases of earthquakes and explosions, which was captured at a seismic recording station.  Another example is the data about returns on the New York Stock Exchange.  See these and many other time series data in the book by Shumway and Stoffer \cite{Shumway} and other books on time series.
\end{remark}

\section{Proofs}\label{proof}

\begin{lemma} \label{lemma1}
	If the conditions of Theorem \ref{thm1} (or \ref{thm2}) hold, then
	\begin{align} \label{lbl2}
	\sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big| = O \bigg( \bigg( \frac{\log n}{n} \bigg) ^ \frac{2}{5} \; \bigg)
	\end{align}
	almost surely.
\end{lemma}

\begin{proof}
	This lemma follows from Theorem 2 of Wu et al. \cite{WHH} (see their discussion immediately after the statement of Theorem 2 and in the penultimate paragraph of section 4.1)  Also, see the discussion in the appendix on fundamental results.

\end{proof}

\begin{lemma} \label{lemma3}
	If the conditions of Theorem \ref{thm1} (or \ref{thm2}) hold, then
	\begin{align} \label{lbl30}
	\gamma_n^5 \gg \frac{\log n}{n}.
	\end{align}
\end{lemma}

\begin{proof}
	Because $h_n\asymp (n^{-1}\log n)^\frac{1}{5}$, there exists $C\in{\mathbb R}^+$ such that
	\[ \frac{h_n^5}{n^{-1} \log n} > C \] for sufficiently large $n$.  Therefore,
	\begin{align*}
	\lim_{n\to\infty} \frac{\gamma_n^5}{n^{-1} \log n}
	& = \lim_{n\to\infty} \frac{\gamma_n^5}{h_n^5} \cdot \frac{h_n^5}{n^{-1} \log n} \\
	& \geq C  \lim_{n \to \infty} \bigg( \frac{\gamma_n}{h_n} \bigg)^5 \to \infty
	\end{align*}
	as $n\to\infty$, from which \ref{lbl30} follows.
\end{proof}

\noindent\textbf{Note.} Our use of Lemma \ref{lemma3} in the proofs of Theorems \ref{thm1} and \ref{thm2} will be tacit.
$\;$\\
\begin{lemma} \label{lemma2}
	If $\nu$ is a finite signed measure that is absolutely continuous with respect to a measure $\mu$, then corresponding to every positive number $\varepsilon$ there is a positive number $\delta$ such that $|\nu|(E)<\varepsilon$ whenever $E$ is a measurable set for which $\mu(E)<\delta$.
\end{lemma}

\begin{proof}
	This is a basic result from measure theory.  See, for example, Theorem B of Halmos \cite{Halmos} in section 30.
\end{proof}

\paragraph{Proof of Theorem \ref{thm1}.} We begin with the decomposition
\begin{align} \label{lbl4}
\begin{split}
S_n(f) -&\int_{A_n} \big(-\log f(x)\big) f(x)\;dx \\
&= -\int_{A_n} f_n(x)\;\log f_n(x)\;dx + \int_{A_n} f(x)\;\log f(x)\;dx \\
&= -\int_{A_n} f_n(x)\;\log f_n(x)\;dx + \int_{A_n} f(x)\;\log f_n(x)\;dx \\
&\ \;\;\; -\int_{A_n} f(x)\;\log f_n(x)\;dx + \int_{A_n} f(x)\;\log f(x)\;dx \\
&= I_{n,1} + I_{n,2},
\end{split}
\end{align}
where
\begin{align*}
I_{n,1} &:= -\int_{A_n} \big[ f_n(x)\; - f(x) \big]\;\log f_n(x)\;dx,
\end{align*}
and
\begin{align*}
I_{n,2} &:= -\int_{A_n} f(x) \; \big[ \log f_n(x)\; - \log f(x) \big]\;dx,
\end{align*}

First, we consider $I_{n,1}$.  Using the inequality
\begin{align*}
|\log z| \leq z + \frac{1}{z}
\end{align*}
for $z\in{\mathbb R}^+$, we notice that for all $x\in A_n$, we have
\begin{align*}
\big|\log f_n(x)\big|
&\leq f_n(x) + \frac{1}{f_n(x)} \\
&= \bigg( 1 + \frac{1}{(f_n(x))^2} \bigg) f_n(x) \\
&\leq \bigg( 1 + \frac{1}{\gamma_n^2} \bigg) f_n(x).
\end{align*}
It follows that
\begin{align} \label{boundIn1}
\begin{split}
\big| I_{n,1} \big|
&\leq \sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big| \int_{A_n} \big|\log f_n(x)\big|\;\;dx \\
&\leq \bigg( 1 + \frac{1}{\gamma_n^2} \bigg) \sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big|,
\end{split}
\end{align}
since $f_n(x)$ integrates to unity over the real line.

Next, we consider $I_{n,2}$.  Since the set over which we are integrating may be changed to $A_n \cap \{x : f(x) > 0\}$ without affecting the value of $I_{n,2}$, we may assume that $f$ is positive on $A_n$.  Using the inequality
\begin{align*}
\log z \leq |z-1| + |z^{-1}-1|
\end{align*}
for $z\in{\mathbb R}^+$, we notice that for all $x\in A_n$, we have
\begin{align} \label{lblineq}
\begin{split}
\big|\log f_n(x)\; - \log f(x) \big|
&= \bigg| \ln \bigg( \frac{f_n(x)}{f(x)} \bigg) \bigg| \\
&\leq \bigg| \frac{f_n(x)}{f(x)} -1 \bigg| + \bigg| \frac{f(x)}{f_n(x)} -1 \bigg| \\
&= \bigg| \frac{f_n(x)-f(x)}{f(x)} \bigg| + \bigg| \frac{f(x)-f_n(x)}{f_n(x)} \bigg| \\
&= \bigg(1+\frac{f_n(x)}{f(x)}\bigg)  \bigg| \frac{f_n(x)-f(x)}{f_n(x)} \bigg| \\
&\leq \frac{C}{\gamma_n}  \big| f_n(x)-f(x) \big|,
\end{split}
\end{align}
if we can justify the existence of $C\in{\mathbb R}^+$.  To that end, define
\begin{align*}
\varepsilon_n = \sup_{x\in A_n} \big| f_n(x)\; - f(x) \big|,
\end{align*}
and note that for all $x\in A_n$, we have
\begin{align*}
\bigg| 1 - \frac{f(x)}{f_n(x)} \bigg| \leq \frac{\varepsilon_n}{f_n(x)} \leq \frac{\varepsilon_n}{\gamma_n}.
\end{align*}
Taking the supremem over $A_n$ yields
\begin{align*}
\sup_{x\in A_n} \bigg| 1 - \frac{f(x)}{f_n(x)} \bigg| &\leq \frac{\varepsilon_n}{\gamma_n} = \gamma_n^{-1}\;\varepsilon_n \\
&\leq C \gamma_n^{-1}\bigg/\bigg(\frac{n}{\log n}\bigg)^{\frac{2}{5}},
\end{align*}
by Lemma \ref{lemma1}.  Note that
\begin{align*}
\gamma_n^{-1}=o\bigg( \bigg( \frac{n}{\log n} \bigg)^\frac{2}{5} \bigg),
\end{align*}
since
\begin{align*}
\lim_{n\to\infty} \frac{\gamma_n^{-1}}{ \big(\frac{n}{\log n}\big)^\frac{2}{5}}
&= \lim_{n\to\infty} \frac{ \big(\frac{\log n}{n}\big)^\frac{2}{5}}{\gamma_n} \\
&= \lim_{n\to\infty} \bigg(\frac{ \frac{\log n}{n}}{\gamma_n^5}\;\frac{ \frac{\log n}{n}}{1} \bigg)^\frac{1}{5} \\
&= 0.
\end{align*}
This guarantees the existence we sought to establish.  We continue with
\begin{align} \label{boundIn2}
\begin{split}
\big| I_{n,2}  \big|
&\leq \frac{C}{\gamma_n}\ \sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big| \int_{A_n} f(x)\;dx \\
&\leq \frac{C}{\gamma_n}\ \sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big|,
\end{split}
\end{align}
since $f_n(x)$ integrates to unity over the real line.

In view of (\ref{lbl4}), (\ref{boundIn1}), and (\ref{boundIn2}), we have shown that
\begin{align} \label{lbl1}
\bigg|S_n(f) - &\int_{A_n} \big(-\log f(x)\big) f(x)\;dx \bigg| \\
&\leq \bigg(\frac{1}{\gamma_n^2} + \frac{C}{\gamma_n} + 1 \bigg) \sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big|.
\end{align}
Therefore,
\begin{align*}
\limsup_{n\to\infty}\;&\bigg(\frac{n\;\gamma_n^5}{\log n}\bigg)^\frac{2}{5}\;\;\bigg|S_n(f) - \int_{A_n} \big(-\log f(x)\big) f(x)\;dx\ \bigg| \\
&\leq \limsup_{n\to\infty}\; \bigg(\frac{n}{\log n}\bigg)^\frac{2}{5}\gamma_n^2 \bigg(\frac{1}{\gamma_n^2} + \frac{C}{\gamma_n} + 1 \bigg) \sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big| \\
&= \limsup_{n\to\infty}\; \big( \gamma_n^2 + C \gamma_n + 1 \big) \bigg(\frac{n}{\log n}\bigg)^\frac{2}{5} \sup_{x\in{\mathbb R}} \big| f_n(x)\; - f(x) \big|,
\end{align*}
where the last expression is constant almost surely by Lemma \ref{lemma1} and since $\gamma_n\to 0$.
{\hfill $\square$ \bigskip}

\paragraph{Proof of Corollary \ref{cor1}.}
By the triangle inequality
\begin{align*}
|S_n(f) - S(f)| \leq J_{n,1} + J_{n,2},
\end{align*}
where
\begin{align*}
J_{n,1} = \bigg|S_n(f) - \int_{A_n} \big(-\log f(x)\big) f(x)\;dx\ \bigg|
\end{align*}
and
\begin{align*}
J_{n,2} = \bigg|\int_{A_n} \big(-\log f(x)\big) f(x)\;dx - S(f) \bigg|.
\end{align*}
Since $J_{n,1}\to 0$ almost surely by Theorem \ref{thm1}, we only need to contend with $J_{n,2}$.  That is, we need to show that
\begin{align} \label{lbl3}
\int_{A_n^\mathsf{c}} f(x) \log f(x)\;dx \to 0
\end{align}
almost surely as $n\to\infty$.

For any Borel measurable set $E$, consider
\begin{align*}
P(E) = \int_E f(x)\;dx,
\end{align*}
and define the signed measure
\begin{align*}
\nu (E) = -\int_E \log f(x)\;dP.
\end{align*}
Since $|S(f)|<\infty$, both $\nu^+$ and $\nu^-$ are finite measures, and thus, $\nu$ is a finite signed measure that is absolutely continuous with respect to $P$.  Because of Lemma \ref{lemma2}, it suffices for us to demonstrate that
\begin{align*}
P(A_n^\mathsf{c}) \to 0
\end{align*}
almost surely.  For any $x\in A_n^\mathsf{c}$, we have $f_n(x) < \gamma_n$.  By Lemma \ref{lemma1}, there exists $C\in{\mathbb R}^+$ such that $f(x) \leq f_n(x) + |f_n(x)-f(x)| < \gamma_n + C \big( \frac{\log n}{n} \big)^{\frac{2}{5}}$ almost surely, and hence, we have shown that $A_n^\mathsf{c}\subseteq B_n$ almost surely, where
\[ B_n := \bigg\{x: f(x) < \gamma_n + C \bigg( \frac{\log n}{n} \bigg)^{\frac{2}{5}} \bigg\}. \]
It is easy to see that
\begin{align*}
0 \leq P(A_n^\mathsf{c}) \leq P(B_n) \to 0
\end{align*}
almost surely, since $\gamma_n + C \big( \frac{\log n}{n} \big)^{\frac{2}{5}} \to 0$ as $n\to\infty $.
{\hfill $\square$ \bigskip}

\paragraph{Proof of Theorem \ref{thm2}.}
We start with
\begin{align*}
\bigg\lVert S_n(f)-&\int_{A_n} \big(-\log f(x)\big) f(x)\;dx\ \bigg\rVert_2 \\
&\leq \bigg\lVert S_n(f) + \int_{A_n} f_n(x) \log f(x)\;dx \bigg\rVert_2 \\
&\;\;\;\;\;+ \bigg\lVert \int_{A_n} f_n(x) \log f(x)\;dx - \int_{A_n} f(x) \log f(x)\;dx \bigg\rVert_2 \\
& =: K_{n,1} + K_{n,2}.
\end{align*}
Recall inequality (\ref{lblineq}) in the proof of Theorem \ref{thm1}.  Arguing in a similar manner as before, we can demonstrate the existence of $C_1\in{\mathbb R}^+$ so that
\begin{align*}
K_{n,1} &= \bigg\lVert \int_{A_n} f_n(x)\big[\log f(x) - \log f_n(x)\big]\;dx \bigg\rVert_2 \\
&= \bigg\lVert \int_{A_n} f_n(x) \log \frac{f(x)}{f_n(x)}\;dx \bigg\rVert_2 \\
& \leq \bigg\lVert \int_{A_n} f_n(x) \bigg| \log \frac{f(x)}{f_n(x)}\bigg|\;dx \bigg\rVert_2 \\
& \leq \bigg\lVert \int_{A_n} \frac{C_1}{\gamma_n} |f_n(x)-f(x)| f_n(x) \;dx \bigg\rVert_2\\
& \leq \frac{C_1}{\gamma_n} \; \bigg( \frac{\log n}{n} \bigg)^\frac{2}{5} \bigg\lVert  \int_{x\in A_n}\;f_n(x)\;dx \bigg\rVert_2\\
& \leq \frac{C_1}{\gamma_n} \; \bigg( \frac{\log n}{n} \bigg)^\frac{2}{5}.
\end{align*}
Also, notice that
\begin{align*}
K_{n,2} &= \bigg\lVert \int_{A_n} [f_n(x) - f(x)] \log f(x)\;dx \bigg\rVert_2 \\
&\leq \bigg\lVert \int_{A_n} \big|f_n(x)-f(x)\big| \log f(x)\;dx\bigg\rVert_2 \\
&\leq C_2 \bigg(\frac{\log n}{n}\bigg)^\frac{2}{5} \bigg\lVert \int_{A_n} \bigg( f(x)+\frac{1}{f(x)} \bigg) \;dx\bigg\rVert_2\\
&\leq C_2 \bigg(\frac{\log n}{n}\bigg)^\frac{2}{5} \bigg[1+ \bigg\lVert \int_{A_n} \frac{f_n(x)}{f(x)}\frac{1}{f_n^2(x)}f_n(x)\;dx\bigg\rVert_2\bigg]\\
&\leq C_2 \bigg(\frac{\log n}{n}\bigg)^\frac{2}{5} \bigg(1+ \frac{C_3}{\gamma_n^2}\bigg).
\end{align*}
Therefore,
\begin{align*}
\bigg(\frac{n\;\gamma_n^5}{\log n}\bigg)^\frac{2}{5}&\bigg\lVert S_n(f) - \int_{A_n} \big(-\log f(x)\big) f(x)\;dx\ \bigg\rVert_2 \\
& \leq \bigg(\frac{n}{\log n}\bigg)^\frac{2}{5}\gamma_n^2\;\bigg[\frac{C_1}{\gamma_n}\bigg(\frac{\log n}{n}\bigg)^\frac{2}{5} + C_2 \bigg(\frac{\log n}{n}\bigg)^\frac{2}{5}\bigg(1+\frac{C_3}{\gamma_n^2}\bigg)\bigg] \\
&= C_1 \gamma_n + C_2 \gamma_n^2 + C_2 C_3,
\end{align*}
from which the result follows.
{\hfill $\square$ \bigskip}

\paragraph{Proof of Corollary \ref{cor2}.}
Note the decomposition
\begin{align*}
\sqrt{\textup{MSE}(S_n(f))}&=\big\lVert S_n(f)-S(f) \big\rVert_2 \\
&\leq \bigg\lVert S_n(f) - \int_{A_n} \big(-\log f(x)\big) f(x)\;dx\ \bigg\rVert_2 \\
&\;+  \bigg\lVert \int_{A_n} \big(-\log f(x)\big) f(x)\;dx - S(f) \bigg\rVert_2 \\
&=: M_{n,1} + M_{n,2}.
\end{align*}
By Theorem \ref{thm2}, $M_{n,1}\to 0$.  Now, let
\begin{align*}
W_n = \int_{x\in{\mathbb R}} \int_{y\in{\mathbb R}} f(x) f(y) \log f(x) \log f(y)\;{\mathcal{I}}\big((x,y)\in A_n^\mathsf{c}\times A_n^\mathsf{c}\big)\;dx\;dy
\end{align*}
and
\begin{align*}
W = \int_{x\in{\mathbb R}} \int_{y\in{\mathbb R}} f(x) f(y) \big| \log f(x) \log f(y) \big| \;dx\;dy.
\end{align*}
Recall from (\ref{lbl3}) in the proof of Corollary \ref{cor1} that $W_n \to 0$ almost surely.  Because $|S(f)|<\infty$, it follows that
$W<\infty$, and moreover, $|W_n| \leq W$. Hence,
\begin{align}
\begin{split}
M_{n,2}^2&= \bigg\lVert  \int_{A_n^{\mathsf{c}}} f(x) \log f(x) \;dx\ \bigg\rVert_2^2 \\
&= \bigg\lVert \int_{{\mathbb R}} f(x) \log f(x)\;{\mathcal{I}}(x\in A_n^\mathsf{c})\;dx \bigg\rVert_2^2 \\
&= {{\mathbb E}\,} [W_n],
\end{split}
\end{align}
and the Lebesgue Dominated Convergence Theorem guarantees that
\begin{align*}
\lim_{n\to\infty} M_{n,2}^2 = \lim_{n\to\infty} {{\mathbb E}\,} [W_n] = {{\mathbb E}\,} \bigg[\lim_{n\to\infty} W_n \bigg] = {{\mathbb E}\,} [0] = 0,
\end{align*}
thereby proving the corollary.
{\hfill $\square$ \bigskip}

\section{Appendix}

In the paper \cite{WHH}, Wu et al. establish results that are very useful in the proof section.  Here, we briefly survey their definitions and results which show that the kernel density estimator for one-sided linear processes enjoys properties similar to the independent case---see Stute \cite{Stute}. Their work identifies conditions under which the kernel density estimator enjoys strong uniform consistency for a wide class of time series.  Included is the linear process in (\ref{lp}).

As is common in analysis of time series, we allude to an independent and identically distributed collection $\{\varepsilon_i : i\in{\mathbb Z}\}$ of random variables, typically referred to as the innovations.  Note that many time series models fit the form
\begin{align} \label{J_filter}
X_n = J(\cdots,\varepsilon_{n-1},\varepsilon_{n}),
\end{align}
which regards the $X_n$ as a system dependent on the innovations.  Note here that $J$ is some measurable function which is referred to as the filter.  In this context, we also need to define the sigma algebras
\begin{align*}
\mathcal{F}_n=\sigma\{\varepsilon_{n}, \varepsilon_{n-1}, \cdots\},
\end{align*}
where $n\in{\mathbb Z}$.  In addition. let $\varepsilon_{0}'$ be an independent and identical copy of $\varepsilon_{0}$ which is, of course, independent of all the $\varepsilon_{i}$.  For $n\geq0$, define
\[
\mathcal{F}_n^*=\sigma\{\varepsilon_{n}, \varepsilon_{n-1},\cdots,\varepsilon_{1}, \varepsilon_{0}', \varepsilon_{-1},\cdots \},
\]
and for $n<0$, put $\mathcal{F}_n^*=\mathcal{F}_n$.

Define the $l$-step ahead conditional distribution by
\[
F_l(x|\mathcal{F}_k)=P(X_{l+k} \leq x | \mathcal{F}_k),
\]
where $l\in{\mathbb N}$ and $k\in{\mathbb Z}$.  When it exists, the $l$-step ahead conditional density is
\[
f_l(x|\mathcal{F}_k)=\frac{d}{dx} F_l(x|\mathcal{F}_k).
\]
As Wu et al. \cite{WHH} notes, a sufficient condition for the existence of a marginal density of (\ref{J_filter}) is that $f_1(x|{\cal F}_0)$ exists and is uniformly bounded almost surely by some $M\in{\mathbb R}^+$.  We shall refer to this as the \textbf{marginal condition}.  Similarly, $F_l(x|{\cal F}_k^*)=P(X_{l+k}^*\leq x|{\cal F}_k^*)$, where $X_{l+k}^*=X_{l+k}-a_{l+k}\varepsilon_{0} + a_{l+k}\varepsilon'_{0}$ if $l+k\geq 0$ and $X_{l+k}^*=X_{l+k}$ if $l+k<0$.  Also, $f_l(x|{\cal F}^*)=\frac{d}{dx}F_l(x|{\cal F}_k^*)$.

With this setup, the authors introduce the following measures of the dependence present in the system (\ref{J_filter}).  Now, for $k\geq 0$, define a pointwise measure of difference by
\begin{align*}
\theta_k(x)=\lVert f_{1+k}(x|\mathcal{F}_0)-f_{1+k}(x|\mathcal{F}_0^*)\rVert_2
\end{align*}
and an ${\mathcal{L}}^2$-integral measure of difference over ${\mathbb R}$ by
\begin{align*}
\theta(k)=\bigg[\int_{\mathbb R}\theta_k^2(x)\;dx\bigg]^\frac{1}{2}.
\end{align*}
Finally, define an overall measure of difference by
\begin{align*}
\Theta(n)=\sum_{j\in{\mathbb Z}} \bigg(\sum_{k=1-j}^{n-j}\big|\theta(k)\big|\bigg)^2.
\end{align*}
The distances on the derivatives are defined similarly, as given below.
\begin{align*}
\psi_k(x)&=\lVert f_{1+k}'(x|\mathcal{F}_0)-f_{1+k}'(x|\mathcal{F}_0^*)\rVert_2, \\
\psi(k)&=\bigg[\int_{\mathbb R}\psi_k^2(x)\;dx\bigg]^\frac{1}{2}, \;\;\;\text{ and}\\
\Psi(n)&=\sum_{j\in{\mathbb Z}} \bigg(\sum_{k=1-j}^{n-j}\big|\psi(k)\big|\bigg)^2.
\end{align*}

\noindent With this setup, we can now report the following result of Wu et al. \cite[Theorem 2]{WHH}.

\begin{theorem}
	Assume that, for some positive $r$ and $s$, we have that $K\in\mathcal{C}^r$ is a bounded function with bounded support and that $X_n\in{\mathcal{L}}^s$.  Further, assume the marginal condition, and assume that $ \Theta(n) + \Psi(n)=O(n^\alpha\; \tilde{l}(n))$, where $\alpha \geq 1$ and where $\tilde{l}$ is a slowly varying function.  If  $\log n=o(n h_n)$, then
	\begin{align*}
	\sup_{x\in{\mathbb R}} \big|f_n(x)-{{\mathbb E}\,} f_n(x)\big|=O\bigg(\sqrt{\frac{\log n}{n h_n}} + n^{-\frac{1}{2}}\;l(n)\bigg),
	\end{align*}
	where $l(n)$ is another slowly-varying function.
\end{theorem}

Now consider our particular case when the filter is the linear process of (\ref{lp}).  In view of our assumption that the innovations have finite variance and because we assume the coefficients are square summable, $X_n\in {\mathcal{L}}^2$.  Moreover, we assume all of the bandwidth, kernel, and density conditions listed earlier, from which it easily follows that the marginal condition is satisfied.  For the short memory linear process (under the aforementioned assumptions), Wu et al. \cite{WHH} demonstrated that $\Theta(n) + \Psi(n) = O(n)$.  Also, notice that condition \textbf{B.1} implies that $\log n = o(n h_n)$.  Therefore, the theorem of Wu et al. \cite{WHH} applies to (\ref{lp}).

In addition, the well-known Taylor series argument under the conditions \textbf{K.2} and \textbf{K.3}, as well as \textbf{D.3}, yields
\begin{align*}
\sup_{x\in{\mathbb R}} \big|{{\mathbb E}\,}[f_n(x)]-f(x)\big|=O(h_n^2),
\end{align*}
so collectively, we see that
\begin{align*}
\sup_{x\in{\mathbb R}} \big|f_n(x)-f(x)\big|=O\bigg(\sqrt{\frac{\log n}{n h_n}} + n^{-\frac{1}{2}}\;l(n)+h_n^2\bigg).
\end{align*}
Basic methods of differential calculus show that $\sqrt{\frac{\log n}{n h_n}} + h_n^2$ is minimized when $h_n$ satisfies \textbf{B.1}.  Indeed, the optimum value of  $h_n$ has the exact order of  $\bigg(\frac{\log n}{n} \bigg)^\frac{1}{5}$.

\textbf{Acknowledgement}
The authors are grateful to the referees and Professor Daniel J. Henderson for carefully reading the paper and for insightful suggestions that significantly
improved the presentation of the paper.
The research is supported in part by
the Simons Foundation Grant 586789 and the College of Liberal Arts
Faculty Grants for Research and Creative Achievement at the University of Mississippi.
\newpage

redpajama_set_name: RedPajamaArXiv


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