text:
Electric vehicles may well transform mobility, and thereby the society mobility supports. Professor Yoichi Hori of the University of Tokyo Graduate School talked with Satoshi Sawamura, President of automotive semiconductor leader ROHM Corp., about technical trends in EVs and concepts about the future of mobility.
Next-Gen Technology Driving Vehicle Evolution
Pushing the Merits of Key Technologies
The development of Electric Vehicles (EV) created new demand for automotive semiconductor devices, and continues to be a key growth vector for ROHM. The company was born in 1954 as a manufacturer of resistors, and we began our shift into the semiconductor business in the 1960s. Semiconductor devices were originally primarily for the consumer electronics sector, but from about a decade ago we have been concentrating on growth in automotive and industrial equipment sectors. Today, these two sectors account for about 43% of sales revenue. EVs represent only a fairly small portion of the total automotive market at present, but they are expected to achieve significant growth in the near future.
If fundamental, significant evolution in EVs can be achieved, they will spread extremely rapidly, I think.
EVs today are still constrained by the concepts of traditional automobiles, basically just replacing the conventional automobile engine with an electric motor. EVs have a number of characteristics that are not found in engine automobiles, however, specifically torque response is much faster, the power train can be distributed, and generated torque can be measured accurately. I believe that an EV that effectively leverages these characteristics will offer unique added value. For example, the precise control afforded by motor drive minimizes tire slip, which means that tires can be made narrower without losing any mobility, improving mileage.
Each time EVs demonstrate significant evolution, I think the EV market will experience a major spurt of growth.
Electrification is progressing must faster than the industry expected, but even so I don't expect the automotive market to suddenly switch over to EVs. I think conventional automobiles will remain on the market for some time. That's why our automotive semiconductor business is based on a gradual shift from engine designs to EVs, and then to autonomous driving.
Fig.1 The ROHM Solution for Electric Vehicles
The development of new automotive technologies is accelerating, and the media is covering the industry more frequently as a result. There seems to be a lot more interest in when EVs will replace engine designs, and when autonomous driving will finally enter widespread use. I agree that it's unlikely to change rapidly, though: there will probably be a long period when the technologies coexist. I think it's a good idea to grow your business in line with market reality.
Our automotive semiconductor business is designed to address three major themes: safety, automation and Advanced Driver Assistance Systems (ADAS), and electrification and energy efficiency. As part of our approach to safety we strive to deliver zero defects, delivering the highest possible quality in our entire range of automotive products from high-performance large-scale integration (LSI) chips down to the general-purpose components such as resistors and discretes that we command such a large market share in.
We are also working on developing new products offering improved functional safety. For example, the instrument clusters and side mirrors in cars are beginning to transition to electronic designs, and faults in their electronic systems can lead to major accidents. ROHM was the first in the industry to offer a liquid crystal display (LCD) chipset for cluster panels with onboard functions to detect chip errors and faults, helping prevent accidents from happening. We implemented the technology in this product first because we recognized the high degree of functional safety required in automotive electronics.
As automotive electronics becomes increasingly pervasive, electronics will become even more critical in assuring safety. The role of semiconductor manufacturers will become even more important. What about the other two themes you mentioned?
We are supporting development in automation and ADAS primarily through the communications integrated circuits (IC) that empower sensing devices, and safe power supply.We are supporting development in automation and ADAS primarily through the communications integrated circuits (IC) that empower sensing devices, and safe power supply.
Our activity in automobile electrification and energy efficiency covers a wide range of vehicle types, not only EVs. The 48V mild hybrids gaining popularity in Europe, for example, use a state-of-the-art power supply IC featuring ROHM's proprietary "nano-pulse control" high-efficiency supply technology. For inverters in plug-in hybrids and EVs we provide silicon carbide (SiC) power devices, gate drivers, and other chips.
As a university researcher I've been studying the evolution of automobiles over a long span. I'm specifically interested in applications such as EV control, wireless power transmission systems, and electric double-layer capacitors, or supercapacitors, capable of rapid charge and discharge. The underlying premise to my work is that the vehicles of a century from now will run on motors, capacitors, and wireless technology.
It's pretty clear that vehicle drive power is changing from engines and motors, but I don't think the EVs of today will achieve widespread market penetration. One of the reasons is problems with energy distribution. EVs today store energy in onboard Li-ion rechargeable batteries, and the vehicle can't be used for long periods of time while charging. Worse, the batteries can only store relatively small amounts of energy… in fact, the amount of energy stored in the batteries of an EV is a factor of ten less than the energy of a single tank of gasoline. As a result, EV drivers have to always be careful of how far they can get on their remaining charge
It would be best, in my opinion, to have an infrastructure like that of electric trains, supplying EVs directly with power. Even long-haul drives would be quite possible if you have quick-charge systems operating while waiting at red lights or toll gates, combined with slow-charging technology to gradually charge vehicles via roadway charging systems. There would be no need for a single vehicle to store so much energy. If that sort of infrastructure could be implemented it would revolutionize mobility and society. Any system supplying power from infrastructure, however, will depend on capacitors capable of rapid charge and discharge, and wireless power supply technology to connect vehicles to the electric power grid.
A number of advanced technologies will need to be developed as a base to get that kind of futuristic system kickstarted: we call them "seed technologies." These seed technologies can "bloom" in the form of new ideas and systems, and the technologies to implement them. By rapidly providing semiconductor devices with outstanding characteristics and function, we can contribute to the continuing evolution of the automobile.
Yes, semiconductor devices are unquestionably seed technologies. In fact, I believe that the practical wireless power supply system, offering non-contact electric power transmission, was first achieved using high-efficiency SiC power devices.
The team led by Associate Professor Hiroshi Fujimoto of the University of Tokyo Graduate School of Frontier Sciences, for example, prototyped an EV using in-wheel motors powered by wireless transmission in May 2015, in collaboration with two Japanese firms, Toyo Denki Seizo KK and NSK Ltd. The drive motors are mounted inside each of the four wheels. In the past motor power was supplied through cables, but the cables were constantly subject to load as the vehicle bounced up and down. The prototype supplies wireless power through electromagnetic resonant coupling, eliminating the need for cables.
That wireless power supply system attained a high transmission efficiency (ratio of transmitted to received power) of 94.3%, thanks to SiC power devices. The system converts the direct current (DC) power required by the drive motors into high-frequency alternating current (AC) for wireless transmission, and a key point in achieving the high transmission efficiency is reducing losses generated in the power conversion circuit. The in-wheel motors also use high-efficiency SiC power devices to further minimize conversion losses. As a whole, the system improves power transmission efficiency to the level of practical utilization.
In March 2017, the same research group announced another prototype vehicle using a wireless power supply system, but the power was supplied to the in-wheel motors by transmitters located in the roadway itself (Fig. 2). This second-generation design also achieved a power transmission efficiency of over 90% through the use of SiC power devices in the wireless transmission system, the step-up chopper, the motor control inverter, and other electronics.
Fig.2 Second-Generation In-Wheel Motor EV Developed by Professor Fujimoto of the University of Tokyo Graduate School
SiC power devices are one of the products we are prioritizing. Compared to the silicon (Si) that has been the most common semiconductor material, SiC is a next-gen material offering superior electrical properties. SiC power devices can contribute significantly to reducing energy consumption and power levels in a range of relatively high-power circuits. The potential advantages of SiC power devices, however, are not as widely known as they could be. As part of our effort to improve the situation, in 2016 we began serving as an official technology partner to the Venturi Formula E racing team. Formula E racing is the most important race for the electric vehicle class.
It is a sort of test bed for advanced technology relating to EVs. For Season 3 (2016/2017) ROHM provided the SiC Schottky barrier diodes for the drive system inverters, and in Season 4 (2017/2018) and the following Season 5 we are also providing SiC MOSFETs to complete the transition to a completely SiC configuration for the inverters. Compared to the partial-SiC inverters used in Season 2, these new full-SiC designs are 43% smaller by volume, with weight reduced by 6 kg (Fig. 3). The weight and size reductions should improve vehicle performance. We're really looking forward to the races!
Fig.3 The Venturi Formula E Team Vehicle and Inverter Comparisons (SiC power device wafer in inset)
I also think it is important to demonstrate superior technology where it can be appreciated by many people, because it may stimulate new and exciting ideas from universities, or other organizations. Inverters and their component semiconductor devices have a crucial role to play in vehicle electrification, as is evident in the wireless power supply application I mentioned earlier. As systems evolve, however, sub-systems such as these can get overlooked. The industry is moving toward tighter integration of electronics and machinery when it comes to motors, but that trend will also mean that people will have fewer opportunities to appreciate the importance of semiconductor devices. I hope that you will continue to display the advantages of new semiconductor device technology in a diverse range of situations.
Changes in society and technological progress are dramatically changing the concept of mobility, and ROHM believes that this period of transition offers us an abundance of opportunities to contribute through technology and products. We're committed to continuing our technical leadership in the automotive sector.


Sonitrol has provided comprehensive school security solutions for the educational community for nearly 50 years. Today, Sonitrol is one of the largest providers of school security systems in North America, serving more than 12,000 schools. We are experts at assessing your school's vulnerabilities and recommending appropriate, cost-effective solutions to secure your facilities.


Nowadays, the Polar Plunge takes place on the K Street side of the Curley Community Center. It is overseen and organized by the L Street Brownies, Center Director Freddy Ahern and his staff. Between 500 and 1,000 people take part each year. More than 500 plunged this year. Promptly at 8:15 a.m., ahern announced the prevailing temperatures: 46 degrees in the water, 38 degrees on the beach. He then cued the countdown – "ten, nine, eight, …" and Tommy blew the bugle call "Charge" on his baritone horn. And then they plunged.
To you newcomers, South Boston's Polar Plunge is free (unless you choose to make a contribution of some kind). It's a fabulous "Southie time," an absolutely unique event nationwide, and open to anyone who wants to take the plunge. See you next year, we hope.


A world renowned religious scholar and leader for millions of American Muslims, Imam Wallace Deen Mohammed (R) dedicated his life to building bridges of interfaith respect and racial harmony. The son of Nation of Islam leader the Honorable Elijah Muhammad, the Imam led the transition of "so-called" Black Muslims and the Nation of Islam to true Islam, dispelling false notions of race and religion to make his community a new world leader for the proper understanding and practice of Al-Islam.
Author, scholar, businessman, and advisor to Presidents, Imam Mohammed founded and presided over The Mosque Cares and Collective Purchasing Conference. He served as an International President for the World Conference on Religion and Peace and on the World Supreme Council of Masajid (Mosques). He was the first Muslim in U.S. history to deliver an invocation on the floor of the U.S. Senate. He inherited the struggle for dignity and economic justice through the African-American saga and Islamic influence woven into the history of blacks both before and after American slavery.
The Imam proclaimed the message of Al-Islam to save his Community and the world. To do so, he had to stand courageously firm against the dogma of the Nation of Islam. Like King or Mandela, he had been able to see over the mountain top of obstacles and difficulties, as he moved strategically to evolve his Community from charismatic leadership and centralized hierarchy to institutional leadership and respect for the group intellect. He removed the trappings of riches and a so-called royal birthright to stand against even members of his family to safeguard his community's moral and material interests.
His religion was Al-Islam; his faith was in G-d and in the nobility of the common man; upon that foundation he built a new house to replace a fallen temple. The Imam proclaimed One G-d and one humanity, and that the Final Prophet sent by the Lord of Abraham, Moses, and Jesus is Muhammed to whom the Qur'an was revealed over 14 centuries ago (Peace and Blessings be upon them all).
Imam Mohammed's mission and vision, emphasizing religious scholarship, education, economic development, and the promotion of universal human excellence live on and continue to transpose a legacy of leadership onto the world stage.
Click here to learn more about the life, leadership, and legacy of Imam W. Deen Mohammed and connect with other institutions inspired by his vision.


Today's show featured an interview with Sandra Moran – International Human Rights Activist, Musician, feminist and prison rights advocate. (begins at about 14 min mark) Over the past 3 decades, she's played an important role in peace negotiations in Guatemala and its women's, cultural and prison rights movements. Sandra joined the Guatemalan human rights movement at 14 and during the 1980s became involved with Guatemala's renowned rebel music group Kin Lalat. Her human rights and musical activism made her a target for death squads and by the late 1980s, she was forced into exile, first in Nicaragua, then Mexico and finally to Vancouver, Canada. In 1994, she returned to Guatemala City to co-found the Women's Sector, an alliance of 33 women's groups that represented Guatemalan women during peace negotiations that ended Guatemala's 36-year civil war in late 1996.As a representative of the Women's Sector, Sandra was a founder of the National Women's Forum, which developed public policies on women's rights in the post-war period. In 1995, Sandra also founded "We Are Women", Guatemala's first lesbian collective and was a central organizer of Guatemala's first gay pride parade.In 2006, she co-founded Casa Artesana (Guatemala City) as an alternative meeting place and artistic space for women and other groups facing discrimination. With funding from CoDev and Vancouver's Hawthorne Foundation, Casa Artesana works to improve living conditions for women in Guatemalan prisons and their children, and provides rights training and advocacy support for women prisoners.
Sandra continues to perform percussion and poetry, and in 2011 helped form the band "Centro-America Feminista," made up of prominent women singer-songwriters from several Central American countries.


The Fisiocomputer LTS-60 – Power Scanning Laser is an apparatus for the treatment of the patient by means of a LASER beam, exclusively therapeutic purposes .
It should be noted that the laser beam ("Light Amplification by Stimulated Emission of Radiation", meaning " Light Amplification by Stimulated Emission of Radiation ") is essentially a light beam, whose main features lie in the monochromatic (like saying that is emitted only one light frequency, unlike the light we are used which is always broad spectrum) and in the geometric consistency (ie, just thanks to the previous feature , you can generate laser beams with angles of opening very small, typically for medical applications in the order of one or two mRad).
From these two characteristics derives the fact that with a laser light is able to easily obtain power density much greater than with traditional light sources, and this fact, in addition to being used in various fields industrali, space and military, is several years also used in the medical field, albeit in different application fields.
It stands out, in fact, the so-called "power- laser " (which mainly generates effects of photo- evaporation and photo -chemical reactions: an example is the laser scalpel) from "mid- laser " and " soft laser" (who exclusively bio stimulating effect).
The other versions , however, (by allaLTS LTS -05 -20) given that employ laser diodes whose emission falls in the infrared, require light guide (visible) that is implemented with the use of a laser diode that emits red light.
Given that these versions have powers of the beam of work considerably higher ("mid- laser"), should always be remembered that , although the mode of action are similar to the version of lower power , these powers the thermal effect is not completely negligible (especially if it adopts the continuous emission with small deflections ), for which particular attention is required by the operator in the dosage of energy (which depends on the combination of percentage of the maximum power, frequency, pulse repetition, scan area of the beam).
The "fisiocomputer LTS" emits, in all its versions, a laser beam scanning, that is deflected automatically by a pair of mirrors placed inside in the vicinity of the laser aperture present posteriorly.
The LTS -00 uses a He-Ne tube that emits a power of 5 mW (continuous in time and not adjustable in intensity) at a wavelength of 633 nM and with a difference of 1 mrad.
The IR versions , however, all use a laser diode that generates the beam – guide at a wavelength of 635 nM, with a power of 5mW.
The beam of the work is instead obtained by an array of laser diodes emitting at the wavelength of 808 nm (infrared). The maximum output power varies from 500 mW CONTINUOUS version of the LTS- 05 to 2 W CONTINUOUS version of the LTS -20. For the laser working can be adjusted both the peak power (from 0 to 100% of nominal) that the emission mode, selectable between continuous and pulsed (in this case from 1 to 5000 Hz , with adjustable duty cycle from 5% to 95 % in steps of 5%).
It should be remembered that the versions LTS- 00 and LTS- 05 are Laser Class 3B.
Instead, the other two versions (LTS -10 and LTS- 20) are Laser Class 4.
For all versions of the beam is deflected automatically both in a "vertical" (ie in the direction of "depth" of the apparatus) in the sense that the "horizontal" (ie in the lateral direction). For each of the two deflections, which are operating simultaneously, are adjustable by the operator is the absolute positioning of the beam scan width.
The sweep speed is fixed to the vertical (and is such that even by setting the maximum amplitude of the human eye perceives a continuous line, and then the resulting frequency is always higher than the frequency of fusion of the retina , generally estimated at 16 Hz) and adjustable for horizontal (this time on much lower values, so that the resulting frequency is always such as to sense the lateral movement of the vertical line). The horizontal scanning speed is adjustable between 0.25 and 2 rad/sec (which is roughly equivalent to a range of horizontal scan times ranging from one to eight seconds for the maximum deflection).
The laser beam can exit the rear opening within a solid angle contained in the horizontal limit of 60 ∞ (+30 ∞ with respect to the central axis) and the vertical limit of 55 ∞ ( from 30 ∞ to 85 ∞ to get up from the vertical to the ground).
Within this solid angle setting limits for deflections ranging from 0 ¬∞ to the theoretical maximum (depending on the opening Laser). If both width adjustments are set to minimum ( 0 ∞), the scan is reduced to a fixed point .


I'm still working on this project. Better pics soon I hope.
1.Place freezer paper shiny side down on the Cricut mat, and pick designs to cut out.
(I used size 3 and pressure 2, speed 3).
2.Peel the design off the Cricut mat slowly and carefully.
3. Iron the freezer paper onto fabric, shiny side down--so it will stick to the fabric.
4.Select paint, paint it on carefully with a paint brush and stay within the lines of the stencil.
5.Let it dry, then peel off the freezer paper.
When I finish this I will post pictures.
I don't have pics of the Cricut part, so I will try to add that as well.
Oh, what kind of paint were you using?
I'm excited to see what you're doing with it!


Cookies assist us enhance your Royal Mail online experience. Now we have to make JMS Transport configuration settings like under. Calix access network options deliver enterprise services in utility-specific packages that do extra, reach farther into the community, and deploy faster. Mississippi Trademark purposes are additionally registered right here.
This service exploits the data hidden in historic information of resolved service tickets, akin to buyer drawback descriptions, service categories and related service resolutions. Interactive Intelligence has more than four,000 customers worldwide. SBL-EAI-05000: Business Service call returned error code JNDI_ERROR and message: Exception: javax.naming.CommunicationException: Cannot find SerialContextProvider; Message: Cannot discover SerialContextProvider.
Earlier than leaving on your trip, contact the BSC for an authorized Worker Parking Type, along with instructions on how one can make the most of the low cost. Its goal is to understand the effectiveness of promoting that generated in-bound calls to Royal Mail related to advertising services provided.
Points with phone/Ethernet service must be reported to Kevin Poliquin (x2-5201). If you do not speak or perceive English well, name the Translating and Decoding Service (TIS) on 13 14 50. Ask them to cellphone Shopper & Business Providers in your behalf.
With a Native Business Service, you may have management over the Ship method (e.g. to EAI JMS Transport), and use exception dealing with (try/catch) to lift an exception ought to the operation fail. Discover out why companies across the UK belief us to ensure their items arrive quickly and safely.


Slip our men's replacement insoles into your favorite pair of footwear for an added cushion of comfort. After logging many miles with you, your hiking boot insoles can become worn and lose some support. Luckily, our men's footbeds are an easy fix to make your old boots like new. Need more assistance for your everyday footwear? Our men's Ortholite® Comfort Foam insoles provide the perfect solution for those with chronic knee and back pain. Extend the life of your Merrell men's footwear today with moisture-wicking, anti-odor, replacement insoles.


Pages/sec Rate of hard page faults.
A page fault occurs when a process requires data that is not in its space in physical memory. If the faulted page is found elsewhere in physical memory, then it is called a soft page fault. A hard page fault occurs when a process requires data that must be retrieved from disk.
The speed of a disk access is in the order of milliseconds, whereas a memory access is in the order of nanoseconds.
Buffer cache hit ratio Percentage of requests served out of buffer cache.
A low value indicates that few requests could be served out of the buffer cache, with the rest of the requests being served from disk. Average Value >= 90% in an OLTP system.
Page Life Expectancy Time page spends in buffer cache without being referenced. A low value means that pages are being removed from the buffer, lowering the efficiency of the cache and indicating the possibility of memory pressure.
Checkpoint Pages/sec Pages written to disk by a checkpoint operation.
Memory Grants Pending Number of processes waiting for memory grant. If this counter value is high, then SQL Server is short of memory.
See sys.dm_ exec_query_memory_grants dmv. Average value = 0.
When database consistency check was performed last time?
Read more about When database consistency check was performed last time?
A quick count of all rows in all tables in a database can be done using sysindexes table, a system table containing one row for each index and table in the database.
The following query shows the rows count for each table in database. List of tables can be found by interrogating sysobjects system table (it contains one row for each object created within a database: constraint, default, log, rule, stored procedure, and so on) . The 'rows' column in the system table 'sysindexes' holds the number of committed rows in the table.
I added indid < 2 condition for restricting the query to looking at the clustered index or heap info, and ignore subsequent indices. Note that only the results of committed transactions are held in the rows column (uncommitted transactions are not).
Reduces network traffic and latency, boosting application performance.
Stored procedure execution plans can be reused, staying cached in SQL Server's memory, reducing server overhead.
Stored procedures help promote code reuse. While this does not directly boost an application's performance, it can boost the productivity of developers by reducing the amount of code required, along with reducing debugging time.
Stored procedures can encapsulate logic. You can change stored procedure code without affecting clients (assuming you keep the parameters the same and don't remove any result sets columns). This saves developer time.
Stored procedures provide better security to your data. If you use stored procedures exclusively, you can remove direct SELECT, INSERT, UPDATE, and DELETE rights from the tables and force developers to use stored procedures as the method for data access.
Business rules in table triggers often expect your application to update the table one row at a time. Also, some triggers generate an error when the code in the trigger assigns to a local variable the value returned by selecting a column from the inserted virtual table. The assignment fails if you're updating multiple rows because the inserted table contains more than one row, so the subquery returns more than a single value.
Recovery to the end of the last full backup, no log backups.
All backup types are permitted, reduced log space for minimal logged, bulk operations, but additional risk exposure is log is damaged and a limit on point-in-time recovery.
All backups types permitted, point-in-time recovery possible.
A row identifier used to lock a single row within a heap.
A single row is locked - index key for a single row in an index.
An 8-kilobyte (KB) page in a database, such as data or index pages.
A lock on an extent (=a contiguous group of 8 pages, such as data or index pages). Used, for example, when an ALTER INDEX REBUILD command is executed on a table and the pages of the table may be moved from an existing extent to a new extent.
It acts like a table-level lock, but on a partition instead of on the table itself.
The entire table, including all data and indexes => blocks all write requests on the entire table (including indexes), it can significantly hurt database concurrency.
The table schema definition is locked.
How do I obtain a Query Execution Plan?
Read more about How do I obtain a Query Execution Plan?


Women ask all the time what fitness routine would burn fat and tone the body in the shortest period of time with the longest-lasting results. Is there an exercise program that is the BEST and that would continue being fun after a few weeks, months or years? Before I answer this question, let me share a little of my background. I have been immensely fortunate to have been … [Read more...] about What is the BEST fitness routine?


text:
\section{Introduction}
Regular variation is often used to describe the tail behavior of random variables.
A measurable function $U:\mathbb R_+\to \mathbb R_+$ is {\it regularly varying} at infinity with index $\rho$, written $U\in RV_{-\rho}$, if
\[
\lim_{t\to\infty} \frac{U(tx)}{U(t)} = x^{-\rho}\,,\mbox{ for all } x>0\,.
\]
When $\rho = 0$ we say $U$ is slowly varying. A random variable $Z$ is regularly varying with index $\rho\geq0$, written $Z\in RV_{-\rho}$, if the tail distribution $\wb F(t) = \mathbb P(Z>t)$ is regularly varying: $\wb F\in RV_{-\rho}$ (see e.g.~\cite{resnick87extreme}). For the sake of simplicity, we only consider non-negative random variables.

In extreme value theory, the notion of regular variation plays an important role in characterizing the domain of attraction of random variables. Namely, a random variable $Z$ is regularly varying with index $\alpha>0$, if and only if
\[
\frac1{\kappa_n}\bigvee_{i=1}^n Z_i\Rightarrow \zeta_\alpha
\]
as $n\to\infty$, where $\indn \kappa$ is some normalizing sequence, $Z_1,Z_2,\dots$ are independent and identically distributed (i.i.d.) copies  of $Z$, `$\Rightarrow$' stands for convergence in distribution and $\zeta_\alpha$ is a standard $\alpha$-Fr\'echet random variable with distribution
\[
\Phi_\alpha(t) = \mathbb P(\zeta_\alpha\leq t) = \exp(-t^{-\alpha})\,, t>0\,.
\]
See e.g.~\cite{resnick87extreme}, Proposition 1.11.
In this case, the random variable $Z$ is said to be in the domain of attraction of $\Phi_\alpha$. The notion of regular variation for multivariate random vectors and stochastic processes have also been investigated extensively. See e.g.~\cite{resnick87extreme,resnick07heavy}, \cite{dehaan06extreme}, \cite{balkema07high}, and \cite{hult05extremal}, among others. Other applications of regular variation include, just to mention a few, the domain of attraction of partial sums of i.i.d.~random vectors (\cite{rvaceva62domains}), large deviations of regularly varying random walks (\cite{hult05functional}) and finite-dimensional distributions of the stationary solution of stochastic recurrence equations (\cite{kesten73random} and \cite{goldie91implicit}).

In this paper, we consider the regular variation of ratios of two random variables $X$ and $Y$. From a practical point of view, the ratio $X/Y$ can be seen as a random normalization of $X$ by $Y$. In extreme value theory, when modeling extremal behaviors, certain normalizations (thresholding) of values are often required. Random normalization sometimes has appealing theoretical properties and simplifies the statistical applications (see e.g.~\cite{heffernan07limit}, Section 4).

By viewing $X/Y$ as the product of $X$ and $1/Y$, the problem is closely related to the regular variation of products of random variables. When the two random variables are independent, this problem has been addressed by \cite{breiman65some}. On the other hand, \cite{maulik02asymptotic} investigated certain dependence cases, which were then applied to the modeling of network traffic.

We address the regular variation of ratios in a specific case. Namely, we suppose that $(X,Y)$ has a {\it bivariate $\alpha$-Fr\'echet distribution} (or, $(X,Y)$ are {\it jointly $\alpha$-Fr\'echet}), i.e., for all $a,b\geq 0$, $\max(aX,bY)$ has $\alpha$-Fr\'echet distribution. To study the bivariate $\alpha$-Fr\'echet distributions, an efficient tool is the {\it spectral representation} introduced by~\cite{dehaan84spectral} and developed by~\cite{stoev06extremal} (a brief review will be given in Section~\ref{sec:preliminaries}). Based on the spectral representation, we provide a necessary and sufficient condition for $X/Y$ to be regularly varying (Theorem~\ref{thm:RV}). If this is the case, then the regular variation index is referred to as the {\it ratio tail index} of $X/Y$. We demonstrate that our condition is easy to check through a few popular models.

Our specific setting provides examples that are not covered by the results in~\cite{breiman65some} and~\cite{maulik02asymptotic}. Furthermore, we show that the ratio tail index does not characterize the dependence between $X$ and $Y$ in the traditional sense. We will compare the ratio tail index and the {\it tail dependence index} (see e.g.~\cite{sibuya60bivariate}, \cite{dehaan77limit} and \cite{ledford96statistics}), which has been widely used to quantify asymptotic (in)dependence of random variables.

As the main application of our result, we derive the asymptotic behavior of the {\it quotient correlation coefficient} (Theorem~\ref{thm:1}) for jointly Fr\'echet distributions. This coefficient was proposed by~\cite{zhang08quotient} and applied to test the independence of two random variables $X$ and $Y$. In this so-called {\it gamma test}, the quotient correlation coefficient is based on the independent samples of ratios $X/Y$ and $Y/X$. The asymptotic behavior of the coefficient has so far been studied only in the case when $X$ and $Y$ are independent Fr\'echet.

Our result provides new theoretical support for the gamma test. We show that, when $(X,Y)$ is bivariate Fr\'echet, the power of the gamma test is high most of the time. Indeed, the asymptotic behavior of the quotient correlation coefficient is essentially determined by the ratio tail indices of $(X,Y)$ and $(Y,X)$, if they exist. Furthermore, if the ratios have lighter tails than the single variables, then the gamma test rejects the null hypothesis with probability going to one as the sample size increases to infinity (Corollary~\ref{coro:power}). We also show that, when the ratios have tails equivalent to the ones of marginals, then in a `worst scenario', the gamma test performs also reasonably well (Example~\ref{example:power}).

The paper is organized as follows. Section~\ref{sec:preliminaries} provides preliminaries on the regular variation and the spectral representations of bivariate $\alpha$-Fr\'echet random vectors. Section~\ref{sec:ratio} is devoted to the characterization of regular variation of $X/Y$, based on the joint distribution of $(X/Y,Y)$.
Section~\ref{sec:examples} calculates the ratio tail indices for several well-known examples.
In Section~\ref{sec:quotientCorrelation}, we review Zhang's gamma test and apply the result in Section~\ref{sec:ratio} to derive the asymptotic behavior of the quotient correlation coefficient in the dependent case. Section~\ref{sec:proofs} provides a proof of the joint distribution of $(X/Y,Y)$, based on a result by~\cite{weintraub91sample}. Finally, in Section~\ref{sec:discussion} we briefly discuss the connection between our results and some related works.

\section{Preliminaries}\label{sec:preliminaries}
In this section, we review the spectral representation of bivariate $\alpha$-Fr\'echet distributions. We also introduce some notations that will be used in the rest of the paper.

We will focus on bivariate 1-Fr\'echet random vector $(X,Y)$.
Every bivariate 1-Fr\'echet random vector $(X,Y)$ has the following spectral representation:
\begin{equation}\label{eq:XY}
(X,Y)\stackrel{\rm d}{=} \bpp{\int^{\!\!\!\!\!\!\!\!\!e}_Sf(s)M({\rm d} s), \int^{\!\!\!\!\!\!\!\!\!e}_Sg(s)M({\rm d} s)}\,.
\end{equation}
Here, `$\ \int^{\!\!\!\!\!\!\!\!\!e}\ \ $' stands for the extremal integral, $(S,{\cal B}_S,\mu)$ is a standard Lebesgue space, for example, a Polish space $(S,\rho)$ with a $\sigma$-finite measure $\mu$ on its Borel sets ${\cal B}_S$, $f,g$ are measurable non-negative and integrable functions on $(S,{\cal B}_S,\mu)$, and $M$ is a 1-Fr\'echet random sup-measure on $(S,{\cal B}_S)$ with control measure $\mu$ (see e.g.~\cite{dehaan84spectral} and \cite{stoev06extremal}). The functions $f$ and $g$ are called the {\it spectral functions} of $X$ and $Y$, respectively, and the joint distribution of $(X,Y)$ can be expressed as follows:
\begin{equation}\label{eq:fdd0}
\mathbb P(X\leq x,Y\leq y) = \exp\bccbb{-\int_S{\frac{f(s)}x\vee \frac{g(s)}y}\mu({\rm d} s)}\,.
\end{equation}
The random vector in~\eqref{eq:XY} is said to be {\it standard}, if it has standard 1-Fr\'echet marginals or equivalently, $\int_S f{\rm d}\mu = \int_S g{\rm d}\mu = 1$. It is well known that any bivariate $\alpha$-Fr\'echet random vector can be easily transformed into a standard 1-Fr\'echet random vector (see e.g.~\cite{stoev06extremal}, Proposition 2.9).

For the spectral representation of standard bivariate 1-Fr\'echet distribution~\eqref{eq:XY}, specific choices of $(f,g)$ and $(S,\mu)$ often appear in the literature (see e.g.~\cite{dehaan06extreme}, Theorem 6.1.14). We will provide examples using the following one for our convenience. Set $f(s) = 2s, g(s) = 2(1-s)$ for $s\in S = [0,1]$, and $\mu = H$, a probability distribution on $[0,1]$ with mean $1/2$. In this case,
\begin{equation}\label{eq:01}
\mathbb P(X\leq x,Y\leq y) = \exp\bpp{-\int_0^{1}\frac{2s}x\vee\frac{2(1-s)}yH({\rm d} s)}\,.
\end{equation}
Many of our examples are constructed by choosing a specific $H$, and their joint cumulative distribution functions often do not have simple forms.

The bivariate $\alpha$-Fr\'echet distributions arise as limits of i.i.d.~bivariate regularly varying random vectors, and it is often convenient to use the notion of {\it vague convergence}, denoted by `$\stackrel{v}{\rightarrow}$' (see e.g.~\cite{kallenberg86random} and~\cite{resnick87extreme}). In particular, a random vector $(V,W)$ is said to be regularly varying with index $\alpha$, if
\begin{equation}\label{eq:VC}
t\mathbb P\bbb{\bpp{\frac V{\kappa_t},\frac W{\kappa_t}}\in\cdot}\stackrel{v}{\longrightarrow} \nu(\cdot)
\end{equation}
in $M_+(\mathbb E)$.
Here, $\kappa_t\in RV_{1/\alpha}$, $\mathbb E = [0,\infty]^2\setminus\{(0,0)\}$, $M_+(\mathbb E)$ is the space of all nonnegative Radon measures on $\mathbb E$, the limit $\nu\in\mathbb E$ is non-zero and $\nu(c\,\cdot) = c^{-\alpha}\nu(\cdot)$, for all $c>0$. The measure $\nu$ is called the {\it exponent measure}, and it is said to have non-degenerate marginals, if $\nu((x,\infty]\times[0,\infty])$ and $\nu([0,\infty]\times(y,\infty])$ are non-degenerate in $x$ and $y$, respectively.
By Proposition 5.11 in \cite{resnick87extreme}, when $\alpha = 1$ and $\nu$ has non-degenerate normalized marginals, one can write
\[
\nu\bccbb{([0,x]\times[0,y])^c} = \int_0^1\frac{2w}x\vee\frac{2(1-w)}y H({\rm d} w)
\]
such that $\int_0^1H({\rm d} w) = \int_0 2wH({\rm d} w) = 1$. In this case, letting $(V_i,W_i), i=1,\dots,n$ be i.i.d.~copies of $(V,W)$, the vague convergence~\eqref{eq:VC} is equivalent to
\begin{equation}\label{eq:VW}
\bpp{\frac1{\kappa_n}\bigvee_{i=1}^n V_i, \frac1{\kappa_n}\bigvee_{i=1}^n W_i} \Rightarrow (X,Y)
\end{equation}
with $(X,Y)$ defined as in~\eqref{eq:01}.

For any random vector $(V,W)$, we say that $V$ and $W$ are {\it asymptotically independent}, if~\eqref{eq:VW} holds with independent $X$ and $Y$. This corresponds to the case when $f$ and $g$ have disjoint supports in~\eqref{eq:XY}, or $H = (\delta_{\{0\}}+\delta_{\{1\}})/2$ in~\eqref{eq:01} (after normalization), or  the exponent measure $\nu$ in~\eqref{eq:VC} concentrates on the two axes $\{(x,0), x>0\}\cup\{(0,y),y>0\}$ with non-degenerate marginals.

Throughout this paper, for all measurable functions $f$ and measurable sets $S_0\subset S$, we write $\nn f_{S_0} = \int_{S_0}|f(s)|\mu({\rm d} s)$, and define
\begin{equation}\label{eq:Dt}
D_t\mathrel{\mathop:}=\{s\in S:f(s)/g(s) \leq t\}\quad\mbox{ and }\quad E_t\mathrel{\mathop:}=  S\setminus D_t\,.
\end{equation}
Here and in the sequel, we will use the convention $1/0 = \infty$.
For any standard bivariate 1-Fr\'echet random vector $(X,Y)$ as in~\eqref{eq:XY}, two important identities are
\[
\lim_{t\to\infty}\nn f_{E_t} = \int_Sf{\bf 1}_{\{g=0\}}{\rm d}\mu\quad\mbox{ and }\quad\lim_{t\to\infty} \nn g_{D_t} = \nn g_S = 1\,,
\]
which follow from the dominated convergence theorem. Furthermore, the joint distribution~\eqref{eq:fdd0} can be expressed as
\begin{equation}\label{eq:fdd}
\mathbb P(X\leq x,Y\leq y) = \exp\bbb{-\bpp{x^{-1}{\nn f_{E_{x/y}}} + y^{-1}\nn g_{D_{x/y}}}}\,.
\end{equation}
Finally, for any random variable $Z$ and $u\geq 0$, we consider a thresholded version of $Z$ denoted by
\begin{equation}\label{eq:u}
 Z(u) = u+(Z-u)_+ = \max(Z,u)
\end{equation}
as in \cite{zhang08quotient}. We consider such thresholded random variables in order to analyze the quotient correlation coefficient (\eqref{eq:qun} below), although it will turn out that the threshold value $u$ does not play an essential role in the asymptotic behavior (see Remark~\ref{rem:CLL} below).
For a random vector $(X,Y)$ in $\mathbb R^2$, let $\{(X_i,Y_i)\}_{i=1,\dots,n}$ denote $n$ i.i.d.~copies of $(X,Y)$.

\section{Ratios of Fr\'echet Random Variables}\label{sec:ratio}
In this section, we provide an explicit formula for the distribution of $X(u)/Y(u)$, $u\geq 0$. Here $X(u) = X$ if $u = 0$. This leads to a necessary and sufficient condition for $X(u)/Y(u)$ to be regularly varying. In the sequel, all the bivariate 1-Fr\'echet random vectors $(X,Y)$ with representation~\eqref{eq:XY} are assumed to be standard. We write $a_t\sim b_t$ if $\lim_{t\to\infty} a_t/b_t = 1$.

Our first result is an explicit formula for the joint distribution of $(X/Y,Y)$.
\begin{Prop}\label{prop:1}
Consider $(X,Y)$ as in~\eqref{eq:XY}. Then, for all $t\geq0, u\geq 0$,
\begin{equation}\label{eq:prop1}
\mathbb P(X/Y\leq t,Y>u)
= \bpp{1+\frac {\nn f_{E_t}}{t\nn g_{D_t}}}^{-1}\bbb{1-\exp\bpp{-\frac{\nn g_{D_t}+t^{-1}{\nn f_{E_t}}}u}}\,.
\end{equation}
In particular, when $\mu(E_t) = 0$, we have $\mathbb P(X/Y>t) = 0$.
\end{Prop}
The proof borrows a result from~\cite{weintraub91sample}, and is deferred to Section~\ref{sec:proofs}.
\begin{Rem}
Proposition~\ref{prop:1} can be seen as a special case of the conditional limit law established in~\cite{heffernan07limit} (Propositions 4 and 5 therein), where $(X,Y)$ are assumed to satisfy certain regular-variation type condition. Their results describe the asymptotic limit of $t\mathbb P[(X/Y,Y/t)\in\cdot]$ as $t\to\infty$. Here, thanks to the spectral representation, we provide an explicit formula for the joint distribution of $(X/Y, Y)$. In particular, our result readily implies that
\begin{eqnarray*}
\lim_{u\to\infty}\mathbb P(X/Y\leq t\mid Y>u) & = &
\lim_{u\to\infty}\frac{\mathbb P(X/Y\leq t, Y>u)}{\mathbb P(Y>u)}
\\
& = & \lim_{u\to\infty}\bpp{1+\frac {\nn f_{E_t}}{t\nn g_{D_t}}}^{-1}\frac{1-\exp[-(\nn g_{D_t}+t^{-1}{\nn f_{E_t}})u^{-1}]}{1-\exp(-u^{-1})}\\
& = & \nn g_{D_t} = \int_{S}g{\bf 1}_{\{f\leq tg\}}{\rm d}\mu\,,
\end{eqnarray*}
which recovers Equation (32) in~\cite{heffernan07limit}. For a more general and geometric treatment of the conditional limit law, see~\cite{balkema07high}.
\end{Rem}
Now, the distribution of $X(u)/Y(u)$ follows as a corollary.
\begin{Coro}
Consider $(X,Y)$ as in~\eqref{eq:XY}. Then, for all $t\geq1, u\geq 0$,
\begin{equation}\label{eq:Xu/Yu}
\mathbb P\bpp{\frac{X(u)}{Y(u)}>t} = \bpp{1+\frac{\nn g_{D_t}}{\nn f_{E_t}}t}^{-1}\bbb{1-\exp\bpp{-\frac{\nn g_{D_t}+t^{-1}{\nn f_{E_t}}}u}}\,.
\end{equation}
In particular, for all $u\geq 0$, as $t\to\infty$,
\begin{equation}\label{eq:Xu/Yu2}
\mathbb P\bpp{\frac{X(u)}{Y(u)}>t} \sim \frac{\nn f_{E_t}}t\bb{1-\exp(-u^{-1})}\,.
\end{equation}
\end{Coro}
\begin{proof}
Observe that for $t\geq1$,
\begin{eqnarray*}
& &
\mathbb P\bpp{\frac{X(u)}{Y(u)}>t} = \mathbb P(X>t(u+(Y-u)^+), X\geq u) \\
& & \ \ \ \  \quad = \mathbb P(X>tu, Y\leq u) + \mathbb P(X>tY, Y>u)\\
& & \ \ \ \  \quad = \mathbb P(Y\leq u) - \mathbb P(X\leq tu, Y\leq u) +\mathbb P(Y>u) - \mathbb P(X\leq tY, Y>u)\\
& & \ \ \ \  \quad = 1 - \mathbb P(X\leq tu, Y\leq u) - \mathbb P(X\leq tY, Y>u)\,.
\end{eqnarray*}
Plugging in~\eqref{eq:fdd} and~\eqref{eq:prop1} yields~\eqref{eq:Xu/Yu}, whence~\eqref{eq:Xu/Yu2} follows immediately, noting $\lim_{t\to\infty} \nn g_{D_t} + t^{-1}\nn f_{E_t} = \nn g_S = 1$.\ifthenelse{\boolean{format}}{\qed}{}
\end{proof}
Inspired by~\eqref{eq:Xu/Yu2}, define
\begin{equation}\label{eq:gamma}
\gamma(t) \mathrel{\mathop:}= \frac{\nn f_{E_t}}t = \frac1t{\int_Sf{\bf 1}_{\{f>tg\}}{\rm d}\mu}\,.
\end{equation}
Clearly, $\lim_{t\to\infty} \gamma(t) = 0$. Then, provided $\gamma(t)>0$, or equivalently, $\mu(E_t)>0$ for all $t\in(0,\infty)$,
~\eqref{eq:Xu/Yu2} implies that for all $x>0$,
\[
\frac{\mathbb P( X(u)/ Y(u)>tx)}{\mathbb P( X(u)/ Y(u)>x)}
\sim \frac{\gamma(tx)/(1+\gamma(tx))}{\gamma(x)/(1+\gamma(x))} \sim \frac{\gamma(tx)}{\gamma(x)}\,, \mbox{ as } t\to\infty\,.
\]
Therefore, we have thus proved the following result.
\begin{Thm}\label{thm:RV}
Consider $(X,Y)$ given in~\eqref{eq:XY} and suppose $\mu(E_t)>0$ for all $t\in(0,\infty)$. Then,
\begin{equation}\label{eq:RV}
X(u)/Y(u)\in RV_{-\alpha} \mbox{ for some (all) } u\geq 0\mbox{, if and only if } \gamma\in RV_{-\alpha}\,.
\end{equation}
\end{Thm}
In other words, for bivariate 1-Fr\'echet random vector $(X,Y)$, to study the regular variation of $X/Y$, it is equivalent to study the regular variation of $\gamma(t)$ in~\eqref{eq:gamma}, based on the spectral functions. We will see in the next section that for many well-known examples, the regular variation of $\gamma(t)$ follows from simple calculations.
\begin{Rem}
When $\mu(E_t) = 0$, or equivalently $\gamma(t) = 0$ for $t$ large enough, we have $\mathbb P(X(u)/Y(u)>t)\leq \mathbb P(X/Y>t) = 0$ by Proposition~\ref{prop:1}.
This situation is relatively simple, and we do not study this case in this paper.
\end{Rem}
\begin{Rem}
A more general setting should be to consider $(X,1/Y)$ in some domain of attractions, such that $X/Y\in RV_{-\alpha}$. \cite{maulik02asymptotic} investigated certain general cases under this framework. Our specific case, however, is not covered by their results. See more discussion in Section~\ref{sec:discussion}.
\end{Rem}
From now on, we say $(X,Y)$ has {\it ratio tail index} $\alpha$, if~\eqref{eq:RV} holds. We immediately have the following consequences.
In the sequel, for any non-decreasing function $U$ on $\mathbb R$, let $U^\leftarrow(y)\mathrel{\mathop:}= \inf\{s:U(s)\geq y\}$ denote the left-continuous inverse of $U$.
\begin{Coro}\label{coro:2}
Consider $(X,Y)$ given in~\eqref{eq:XY} and suppose $\mu(E_t)>0$ for all $t\in (0,\infty)$. Suppose that $(X,Y)$ has ratio tail index $\alpha$, then the following statements hold:
\begin{itemize}
\item[(i)] $\alpha = 1$ if and only if $\nn f_{E_t}$ is slowly varying,
\item[(ii)] for all $u\geq 0$ and for all sequences $\kappa_n\sim (1/\gamma)^\leftarrow(Cn)$,
\begin{equation}\label{eq:convergence}
\frac1{\kappa_n}\bigvee_{i=1}^n\frac{X_i(u)}{Y_i(u)}\Rightarrow \zeta_\alpha\,,
\end{equation}
where $C = 1-\exp(-u^{-1})$ and $\zeta_\alpha$ is a standard $\alpha$-Fr\'echet random variables, and
\item[(iii)] for any sequence $\{u_n\}_{n\in\mathbb N}$ such that $u_n = o(n)$, the convergence in~\eqref{eq:convergence} holds with $u$ replaced by $u_n$ and $\kappa_n\sim(1/\gamma)^\leftarrow(n/u_n)$.
\end{itemize}
\end{Coro}
\begin{proof}
Part (i) is trivial. Part (ii) follows from Proposition 1.11 in \cite{resnick87extreme} and part (iii) can be proved by a similar argument.\ifthenelse{\boolean{format}}{\qed}{}
\end{proof}
An important consequence of Theorem~\ref{thm:RV} is that, the tail of the ratio is always lighter than or equivalent to the tails of $X$ and $Y$. Indeed,~\eqref{eq:gamma} implies $\gamma(t) = O(t^{-1})$ as $t\to\infty$, thus $\gamma\in RV_{-\alpha}$ implies $\alpha\geq 1$.
The case $\alpha = 1$ includes the case when $X$ and $Y$ are independent Fr\'echet random variables. This follows from~\cite{breiman65some}: if $X\in RV_{-1}$, $Z$ has finite $1+\epsilon$ moment for some $\epsilon>0$, and $X$ and $Z$ are independent, then $XZ\in RV_{-1}$ (here, $Z=1/Y$ has standard exponential distribution).

However, for dependent $X$ and $Y$, the ratio tail index $\alpha$ can still equal 1. By Part (i) of Corollary~\ref{coro:2}, a simple case of $\alpha=1$ is when $\lim_{t\to\infty} \nn f_{E_t} = \int_Sf{\bf 1}_{\{g =0\}}{\rm d}\mu>0$. This is the case, in view of the spectral representation~\eqref{eq:01}, when there is a point mass at $\{1\}$, or equivalently when one can write $X = X_1\vee X_2$, such that $X_1$ is 1-Fr\'echet or equal to 0 almost surely, $X_2$ is 1-Fr\'echet, and $X_2$ is independent of $X_1$ and $Y$. If in addition $X_1 = 0$, then $X$ and $Y$ are independent.
We can also have examples such that $\alpha=1$ and $\int_Sf{\bf 1}_{\{g=0\}}{\rm d}\mu = 0$, by constructing $\nn f_{E_t}$ to be slowly varying, in a similar way as in Example~\ref{example:notRV} below.

Next, we compare the ratio tail index and the {\it tail dependence index} defined by
\begin{equation}\label{eq:TDI}
\lambda = \lim_{t\to\infty}\mathbb P(X>t\mid Y>t)\,,
\end{equation}
provided the limit exists. The random variables $X$ and $Y$ are {asymptotically independent} if $\lambda = 0$, and {asymptotically dependent} if $\lambda\in(0,1]$. The tail dependence index has been widely studied and applied. See e.g.~\cite{sibuya60bivariate}, \cite{dehaan77limit}, \cite{ledford96statistics}, among others.

It is easy to see that when $(X,Y)$ is bivariate 1-Fr\'echet, $X$ and $Y$ are asymptotically independent if and only if they are independent, as $(X,Y)$ is in its own domain of attraction.
As a consequence, the tail dependence index (when $\lambda>0$) measures the dependence strength of bivariate 1-Fr\'echet random vectors.
To understand the difference between the tail dependence index and the ratio tail index, observe that in our case,
\[
\lim_{t\to\infty}\mathbb P(X>t\,|\, Y>t) = \nn{\min(f,g)}_S = \int_0^12\min(s,1-s)H({\rm d} s)\,.
\]
The tail dependence index is determined by the spectral measure $H$ on $(0,1)$ (excluding $\{0,1\}$), while the ratio tail index is determined by the behavior of $H$ on a small neighborhood of $\{0,1\}$.
\begin{Example}\label{example:patho}
Consider $(X,Y)$ given by
\begin{equation}\label{eq:rho}
(X,Y) \stackrel{\rm d}{=} ((1-\rho)Z_2\vee \rho Z_3,\rho Z_1\vee (1-\rho)Z_2)\,,
\end{equation}
where $\rho\in[0,1]$ and $Z_1, Z_2$ and $Z_3$ are independent standard 1-Fr\'echet random variables. This corresponds, in~\eqref{eq:01}, to choose
\[
H = \frac\rho2(\delta_{\{0\}}+\delta_{\{1\}})+(1-\rho)\delta_{\{1/2\}}\,,
\]
where $\delta_{\{a\}}$ is the unit point mass at $a$. It is easy to see $\gamma(t) = \rho/t$ for $t>1$. Therefore, $X/Y\in RV_{-1}$, that is, the ratio tail index always equals 1 for $\rho>0$.

Observe that in this model, however, the dependence strength between $X$ and $Y$ varies for different choices of $\rho$. Observe that $\mathbb P(X = Y) = \mathbb P(Z_2\geq \rho/(1-\rho) (Z_1\vee Z_3)) = (1-\rho)/(1+\rho)$, which increases from 0 to 1 as $\rho$ decreases from 1 to 0. By direct calculation, the tail dependence index equals $\lambda = 1-\rho$, which reflects the dependence strength. The ratio tail index does not.
\end{Example}

We conclude this section with a remark on role of the threshold value $u$. The ratio tail indices of several concrete bivariate distributions are calculated in the following section.
\begin{Rem}\label{rem:CLL}
Theorem~\ref{thm:RV} and Corollary~\ref{coro:2} indicate that the threshold value $u$ does not play an essential role in the asymptotic behavior of $X(u)/Y(u)$. Indeed, from~\eqref{eq:convergence}, the limit, if it exists, is independent of $u>0$, up to multiplicative constants. This suggests that letting $u\to\infty$ is more interesting as $n\to\infty$, and part (iii) of Corollary~\ref{coro:2} shows that this would change the convergence rate, but the limit is still the same.

\end{Rem}
\section{Examples}\label{sec:examples}
In this section, we provide several examples on the ratio tail index. The first example is a concrete one where $X/Y$ is not regularly varying.
\begin{Example}\label{example:notRV}
We construct an example based on the spectral representation~\eqref{eq:01}. We will find an $H$ such that $\gamma(t)\sim \exp(-t)$ as $t\to\infty$. Here $\exp(-t)$ is not regularly varying and therefore, by Theorem~\ref{thm:RV}, $X/Y$ is not regularly varying.

Suppose $H$ has Radon--Nikodym derivative $h(s)$ w.r.t.~the Lebesgue measure on $[0,1]$. First, we need to find $h(s)$ such that
\[
\int_{\frac t{1+t}}^12sh(s){\rm d} s = Ct\exp(-t)
\]
for $t$ large enough and some constant $C$ to be chosen later. Solving the above equation, we obtain
\[
h_1(s) = \frac C{2s(1-s)^3}\exp\bpp{-\frac s{1-s}}\,.
\]
One can choose a function $h_0:[0,1/2]\to\mathbb R$ and tune $C$ properly, such that
\begin{equation}\label{eq:H}
H({\rm d} s) = h(s){\rm d} s \quad\mbox{ with }\quad h(s) =\left\{
\begin{array}{l@{\mbox{ if }}l}
h_0(s) & s\in[0,1/2]\\
h_1(s) & s\in[1/2,1]
\end{array}
\right.
\end{equation}
yields a probability measure on $[0,1]$ with mean $1/2$. (For a simple choice, one can consider $H$ to be symmetric by setting $h_0(s) = h_1(1-s)$ for $s\in[0,1/2]$.)
Thus, for $(X,Y)$ corresponding to the spectral representation~\eqref{eq:01} with $H$ given in~\eqref{eq:H},
the extremes of $X$ are compressed by the extremes of $Y$, resulting in an exponentially light tail of $X/Y$, which is not regularly varying.
\end{Example}

The next example is the spectrally discrete bivariate 1-Fr\'echet random vectors.
\begin{Example}\label{example:SD}
A bivariate 1-Fr\'echet random vector $(X,Y)$ is {\it spectrally discrete}, if it has the following representation
\begin{equation}\label{eq:XYSD}
(X,Y)\stackrel{\rm d}{=} \bpp{\bigvee_{i=1}^m a_iZ_i, \bigvee_{i=1}^m b_iZ_i}\,,
\end{equation}
where  $Z_1,\dots,Z_m$ are i.i.d.~standard 1-Fr\'echet random variables, $a_i\geq 0, b_i\geq 0, i = 1,\dots, m$.
This model corresponds, in~\eqref{eq:XY}, to choose $S = \{1,\dots,m\}$, $\mu$ as the counting measure on $S$, $f(i) = a_i$ and $g(i) = b_i$.

Consider~\eqref{eq:XYSD} with $a_i/b_i$ strictly increasing.
If $a_m/b_m<\infty$, then clearly we have $\lim_{m\to\infty}\bigvee_{i=1}^n{X_i}/{Y_i} = {a_m}/{b_m}$ almost surely. Otherwise, if $a_m/b_m = \infty$ ($b_m = 0$) then $\gamma(t) = a_m/t$ for $t>a_{m-1}/b_{m-1}$. Therefore, Theorem~\ref{thm:RV} implies $X/Y\in RV_{-1}$.  That is, the ratio of spectrally discrete vectors is either bounded, or with tail index 1.
\end{Example}
Below, we calculate the ratio tail indices of two popular bivariate Fr\'echet distributions, the {\it logistic model} and the {\it mixed model}. These models have been studied carefully and applied widely to many real data analyses. See \cite{beirlant04statistics} and the references therein for detailed results on these models and their modifications.
\begin{Example}[The logistic model (\cite{gumbel60bivariate})]\label{example:logistic}
Consider the bivariate 1-Fr\'echet distribution
\begin{equation}\label{eq:logistic}
\mathbb P(X\leq x,Y\leq y) = \exp\bbb{-\bpp{\frac1{x^\alpha}+\frac1{y^\alpha}}^{1/\alpha}}\,, x>0, y>0
\end{equation}
for $\alpha\in[1,\infty]$.
Here, the parameter $\alpha$ characterizes the dependence between $X$ and $Y$. The random variables $X$ and $Y$ are independent, if $\alpha$ = 1, and they are fully dependent, i.e., $\mathbb P(X = Y) = 1$, if $\alpha = \infty$.

Example 5.13 in \cite{resnick87extreme} showed that this distribution has a representation~\eqref{eq:XY} with, when $\alpha>1$,
\[
f(s) = (\alpha-1)s^{\alpha-1}(1-s^\alpha)^{-1/\alpha}, g(s) = (\alpha-1)s^{\alpha-2}\,,
\]
and $(S,\mu) = ([0,1],Leb)$.
By straightforward calculation, we have
\[
E_t = \bccbb{s\in[0,1]:\frac{f(s)}{g(s)} > t} = \bccbb{s\in[0,1]:\frac{s^\alpha}{1-s^\alpha} > t^\alpha} = \Big(\frac t{(1+t^\alpha)^{1/\alpha}},1\Big]\,.
\]
Therefore, writing $\wt s = s^\alpha$,
\begin{multline*}
\gamma(t)=\frac{\nn f_{E_t}}
t = \frac1t\int_{\frac t{(1+t^\alpha)^{1/\alpha}}}^1f(s){\rm d} s\\
= \frac1t\int_{\frac{t^\alpha}{1+t^\alpha}}^1\frac{\alpha-1}\alpha(1-\wt s)^{-1/\alpha}{\rm d}\wt s = \frac1t\bpp{\frac1{1+t^\alpha}}^{1-1/\alpha}\sim t^{-\alpha}\,.
\end{multline*}
Theorem~\ref{thm:RV} implies that the logistic model~\eqref{eq:logistic} has ratio tail index $\alpha$. This provides a probabilistic interpretation of $\alpha$. Another way to interpret the parameter $\alpha$ is provided by \cite{ledford98concomitant}. Therein, it is shown that $1/\alpha$ equals the limit probability that the component maxima do not occur at the same observation, i.e.,
\[
\lim_{n\to\infty} \mathbb P({\rm{argmax}}_{i = 1,\dots,n}X_i = {\rm{argmax}}_{i = 1,\dots,n}Y_i) = 1-1/\alpha\,.
\]
\end{Example}
\begin{Rem}[Estimation of the logistic model]
For the logistic model in Example~\ref{example:logistic}, curiously one may want to apply the Hill estimator (\cite{hill75simple}) on i.i.d.~copies $R_i\mathrel{\mathop:}= X_i/Y_i$ to estimate $\alpha$. Let $R_{(1,n)}\leq\cdots\leq R_{(n,n)}$ be the order statistics of $\{R_i\}_{i = 1,\dots, n}$. Then, the Hill estimator is defined by
\[
\what\gamma_H \equiv \what\gamma_H(k,n) \mathrel{\mathop:}=\frac1k\summ i1k \log R_{(i,n)} - \log R_{(k+1,n)}\,,
\]
depending on a threshold integer value $k$. This estimator is consistent in the sense that $\what\gamma_H\to\alpha^{-1}$ in probability, as long as $k/n\to 0$ and $k\to\infty$ as $n\to\infty$ (see e.g.~\cite{dehaan06extreme}~Theorem 3.2.2). Furthermore, the asymptotic normality of $\what\gamma_H$ is guaranteed by the {\it second-order condition} (see e.g.~\cite{dehaan06extreme}, Definition 2.3.1, Theorem~3.2.5). Indeed, since the explicit formula of $\mathbb P(X/Y>t)$ is available, after some calculation we can show
\[
\sqrt k(\what\gamma_H-\alpha^{-1})\Rightarrow{\cal N}(0,\alpha^{-2})\,,
\]
for $k = \left\lfloor n^{\beta/2}\right\rfloor$, $\beta\in(0,2/3)$. When $\beta = 2/3$, the Hill estimator has the optimal rate ($n^{1/3}$), but the limit will be a non-centered normal distribution.

However, applying the Hill estimator here has little practical interest,
since in this parametric model the maximum likelihood estimator works well with better rate ($n^{1/2}$): when $\alpha>1$, the estimation problem is regular and when $\alpha = 1$, the non-regular behavior of the maximum likelihood estimator has been addressed by \cite{tawn88bivariate}. \end{Rem}
\begin{Example}[The mixed model (\cite{gumbel62multivariate})]
Consider the bivariate 1-Fr\'echet distribution
\begin{equation}\label{eq:sibuya}
\mathbb P(X\leq x,Y\leq y) = \exp\bbb{-\bpp{\frac1x+\frac1y-\frac k{x+y}}}, x>0, y>0
\end{equation}
for $k\in[0,1]$. Here, $k = 0$ corresponds to the independent case, though $k = 1$ does not correspond to the full dependence case. By a similar calculation as in Example 5.13 in~\cite{resnick87extreme}, this distribution has a spectral representation~\eqref{eq:01} with
\[
H(\cdot) = k {\rm{Leb}}(\cdot) + \frac{1-k}2[\delta_{\{0\}}(\cdot) + \delta_{\{1\}}(\cdot)]\,,
\]
where $\delta_{\{0\}}$ and $\delta_{\{1\}}$ are unit point masses at $\{0\}$ and $\{1\}$, respectively. Straightforward calculation shows
\[
\gamma(t)= \frac{\nn f_{E_t}}t = \bccbb{{1-k}+k\bbb{1-\bpp{\frac t{1+t}}^2}}\frac1t\,.
\]
Therefore,
\[
\gamma(t)\sim\left\{
\begin{array}{l@{\quad\mbox{ if }\quad}l}
(1-k)t^{-1} & 0<k<1\\
2t^{-2} & k=1\,.
\end{array}
\right.
\]
When $k=1$, the ratio tail index equals $2$, and
\[
\frac1{n^{1/2}}\bigvee_{i=1}^n\frac{X_i}{Y_i}\weakto2^{-1/2}\zeta_{2}\,,
\]
where $\zeta_2$ is standard 2-Fr\'echet. When $k\in(0,1)$, the ratio tail index equals $1$, and
\[
\frac1{n}\bigvee_{i=1}^n\frac{X_i}{Y_i}\Rightarrow\frac1{1-k}\zeta_{1}\,,
\]
where $\zeta_1$ is standard 1-Fr\'echet. The asymptotic behavior of the maxima of the ratios changes dramatically at $k=1$.
\end{Example}

\section{The Quotient Correlation and Independence Test}\label{sec:quotientCorrelation}
In this section, we apply our results on the ratio tail index to study the asymptotic behavior of a test statistic recently proposed by \cite{zhang08quotient}. Therein, Zhang proposed the gamma test for testing the independence between two random variables, based on their ratios.
A similar test was also proposed aiming at testing the asymptotic independence (tail independence). The asymptotic behavior of the test statistics proposed have so far been studied only for independent Fr\'echet random variables. Here, we establish asymptotic results for jointly Fr\'echet random variables with arbitrary dependence structure. We show that, in most of the dependence cases, the power of the hypothesis test goes to one as the sample size increases to infinity.

\cite{zhang08quotient} essentially focused on the test statistics of the form
\begin{equation}\label{eq:qun}
q_{u,n} \mathrel{\mathop:}=\frac{\max_{1\leq i\leq n}( X_i(u)/ Y_i(u)) + \max_{1\leq i\leq n}( Y_i(u)/ X_i(u)) - 2}{\max_{1\leq i\leq n}( X_i(u)/ Y_i(u))\times \max_{1\leq i\leq n}( Y_i(u)/ X_i(u)) - 1}\,,
\end{equation}
where $(X_i(u),Y_i(u))$ are i.i.d.~copies of $(X(u),Y(u))$ (recall~\eqref{eq:u}). When $u=0$, $q_n \mathrel{\mathop:}= q_{0,n}$ is called the {\it quotient correlation coefficient} and when $u>0$, $q_{u,n}$ is called the {\it tail quotient correlation coefficient}.

When $X$ and $Y$ are independent, \cite{zhang08quotient} showed that
\[
\bpp{\frac1n\bigvee_{i=1}^n\frac{ X_i(u)}{ Y_i(u)},\frac1n\bigvee_{i=1}^n\frac{ Y_i(u)}{ X_i(u)}}\Rightarrow\bpp{(1-{\rm e}^{-1/u})\zeta_1\topp1,(1-{\rm e}^{-1/u})\zeta_1\topp2}\,,
\]
and
\begin{equation}\label{eq:nqun}
nq_{u,n}\Rightarrow(1-{\rm e}^{-1/u})^{-1}\bpp{\frac1{\zeta_1\topp1}+\frac1{\zeta_1\topp2}}\stackrel{\rm d}{=} \Gamma(2,(1-{\rm e}^{-1/u})^{-1})\,,
\end{equation}
where $\zeta_1\topp1$ and $\zeta_1\topp2$ are independent standard 1-Fr\'echet random variables. Here $\Gamma(k,\theta)$ stands for the gamma distribution, which equals the distribution of the sum of $k$ independent exponential random variables with mean $\theta$. Recall that the inverse of a standard 1-Fr\'echet random variable has standard exponential distribution.

Based on~\eqref{eq:nqun} with $u = 0$, a hypothesis testing for independence was designed in~\cite{zhang08quotient} with the following null and alternative hypothesis:
\begin{equation}\label{eq:test}
H_0: \mbox{$X$ and $Y$ are independent}\mbox{ and } H_1:\mbox{ $X$ and $Y$ are dependent.}
\end{equation}
The test statistic $nq_{n}$ then has gamma limit distribution as~\eqref{eq:nqun} under the null hypothesis, the test~\eqref{eq:test} is thus referred to as the {\it gamma test}.
When $u > 0$, a similar hypothesis test was designed for testing asymptotic independence.

In this section, we address the asymptotic behavior of $q_{u,n}$ for dependent bivariate 1-Fr\'echet random vectors.
Recall again that for jointly Fr\'echet random variables, the independence and the asymptotic independence are equivalent. Therefore, we will focus on the independence test~\eqref{eq:test} and in particular $q_n = q_{0,n}$.
This essentially requires to investigate the limit of the joint distributions
\[
\bpp{\frac1{\kappa^+_n}\bigvee_{i=1}^n \frac{X_i(u)}{Y_i(u)}, \frac1{\kappa^-_n}\bigvee_{i=1}^n \frac{Y_i(u)}{X_i(u)}}
\]
with $u = 0$ as $n\to\infty$ for some suitable sequence $\indn{\kappa^\pm}$. However, all the asymptotic limits in this section would be the same for all $u>0$, up to multiplicative constants depending on $u$. We choose $u=0$ also for the sake of simplicity.

We first introduce some notations, since we need to deal with two groups of symbols, corresponding to $X/Y$ and $Y/X$ respectively. By default, the symbols with a sign `$+$' (`$-$' resp.) correspond to the ratio $X/Y$ ($Y/X$ resp.). In particular, for $(X,Y)$ as in~\eqref{eq:XY}, write $\gamma_+(t) = \gamma(t)$ as in~\eqref{eq:gamma} and
\[
\gamma_-(t) = \frac{\int_Sg{\bf 1}_{\{g>tf\}}{\rm d}\mu}t\,.
\]
We have shown in Section~\ref{sec:examples} how to calculate the regular variation of $\gamma_+(t)$ for several models. Here $\gamma_-(t)$ can be treated similarly. In particular, if $f(s) = g(1-s)$ for $s\in[0,1]$ and $\mu$ is symmetric on $[0,1]$, or equivalently $H$ in~\eqref{eq:01} is symmetric on $[0,1]$, then $\gamma_+(t) = \gamma_-(t)$.
The following theorem and its corollary generalize Theorems 3.1 and 5.3 in \cite{zhang08quotient}.
\begin{Thm}\label{thm:1}
Consider $(X,Y)$ given by~\eqref{eq:XY}. Suppose $\mu(E_t)>0$ and $\mu(D_t)>0$ for all $t\in(0,\infty)$. Then, $\gamma_\pm\in RV_{-\alpha_\pm}$ for some $\alpha_+>0,\alpha_->0$, if and only if
\begin{equation}\label{eq:asymptoticIndependence}
\bpp{\frac1{\kappa_n^+}\bigvee_{i=1}^n\frac{ X_i}{ Y_i},\frac1{\kappa_n^-}\bigvee_{i=1}^n\frac{ Y_i}{ X_i}}\Rightarrow(\zeta_{\alpha_+},\zeta_{\alpha_-})\,,
\end{equation}
where $\kappa_n^\pm\sim (1/\gamma_\pm)^\leftarrow(n)$ and $\zeta_{\alpha_\pm}$ are independent standard $\alpha_\pm$-Fr\'echet random variables, respectively. When $\alpha_+ = \alpha_- = \alpha$, $\zeta_{\alpha_+}$ and $\zeta_{\alpha_-}$ in~\eqref{eq:asymptoticIndependence} are interpreted as two independent standard $\alpha$-Fr\'echet random variables.
\end{Thm}
\begin{proof}
The `if' part follows from Theorem~\ref{thm:RV}. We show the `only if' part.
Write $R^+ =  X/Y$ and $R^- =  Y/X$.
Observe that, for $t_+, t_->1$,
\[
\mathbb P(R^+\leq t_+, R^-\leq t_-) = 1 - \mathbb P(R^+> t_+) - \mathbb P(R^-> t_-)\,.
\]
Therefore, for all $t_+, t_->0$ and $n$ such that $\kappa_n^+t_+>1$ and $\kappa_n^-t_->1$,
\begin{equation}\label{eq:log}
 \log\mathbb P\bpp{\frac1{\kappa_n^+}\bigvee_{i=1}^n\frac{ X_i}{ Y_i}\leq t_+, \frac1{\kappa_n^-}\bigvee_{i=1}^n\frac{ Y_i}{ X_i}\leq t_-}
\sim -n\bb{\mathbb P(R^+>t_+\kappa_n^+) + \mathbb P(R^->t_-\kappa_n^-)}
\,.
\end{equation}
By definition of $\kappa_n^\pm$, we have
\[
n\mathbb P(R^+>t_+\kappa_n^+)\sim\frac{\mathbb P(R^+>t_+\kappa_n^+)}{\mathbb P(R^+>\kappa_n^+)}\sim t_+^{-\alpha_+}
\]
and similarly $n\mathbb P(R^->t_-\kappa_n^-)\sim t_-^{-\alpha_-}$. Therefore,
\begin{multline*}
\frac{n[\mathbb P(R^+>t_+\kappa_n^+)+\mathbb P(R^->t_-\kappa_n^-)]}{t_+^{-\alpha_+}+t_-^{-\alpha_-}}\\
= \frac{n\mathbb P(R^+>t_+\kappa_n^+)}{t_+^{-\alpha_+}}\frac{t_+^{-\alpha_+}}{t_+^{-\alpha_+}+t_-^{-\alpha_-}} + \frac{n\mathbb P(R^->t_+\kappa_n^-)}{t_-^{-\alpha_-}}\frac{t_-^{-\alpha_-}}{t_+^{-\alpha_+}+t_-^{-\alpha_-}}\to 1
\end{multline*}
as $n\to\infty$. Combined with~\eqref{eq:log} we have thus proved~\eqref{eq:asymptoticIndependence}.\ifthenelse{\boolean{format}}{\qed}{}
\end{proof}
As a corollary, we establish the asymptotic behavior of $q_{n}$. The test statistic $q_n$ has different asymptotic limits, depending on different regular-variation type behaviors of $\gamma_\pm$. We write $a_n = o(b_n)$, if $\lim_{n\to\infty} a_n/b_n = 0$.
\begin{Coro}\label{coro:power}
Suppose that the assumptions in Theorem~\ref{thm:1} hold. Consider $\kappa_n^\pm\sim(1/\gamma_\pm)^\leftarrow(n)$. Then,
\begin{eqnarray}\label{eq:limit}
\kappa_n^+q_n\Rightarrow\zeta_{\alpha_+}^{-1} & \mbox{ if } & \kappa_n^+=o(\kappa_n^-)\,,\nonumber\\
\kappa_n^-q_n\Rightarrow\zeta_{\alpha_-}^{-1} & \mbox{ if } & \kappa_n^-=o(\kappa_n^+)\,,\nonumber\\
\kappa_n^+q_n\Rightarrow\frac1{\zeta_{\alpha_+}}+\frac C{\zeta_{\alpha_-}} & \mbox{ if } & \kappa_n^+\sim C\kappa_n^-\mbox{ for some } C>0\,.\label{eq:kappanqn}
\end{eqnarray}
with $\zeta_{\alpha_\pm}$ as in Theorem~\ref{thm:1}. Note that $\alpha_+>\alpha_-$ implies $\kappa_n^+ = o(\kappa_n^-)$, and $\kappa_n^+\sim C\kappa_n^-$ implies $\alpha_+=\alpha_-$.
\end{Coro}
\begin{proof}
We only prove~\eqref{eq:kappanqn}. The proofs for the other two cases are similar. For the sake of simplicity, write
\begin{equation}\label{eq:Rn}
R_n^+ = \bigvee_{i=1}^n \frac{X_i}{Y_i},\quad R_n^- = \bigvee_{i=1}^n \frac{Y_i}{X_i}
\end{equation}
and $\what R_n^\pm = R_n^\pm/\kappa_n^+$. Then, by~\eqref{eq:qun}, we have
\begin{equation}\label{eq:kappan+qn}
\kappa_n^+q_n = \frac{\what R_n^+ + \what R_n^- - 2/\kappa_n^+}{\what R_n^+ \what R_n^- - 1/(\kappa_n^+)^2}\,.
\end{equation}
Now, Theorem~\ref{thm:1} implies $(\what R_n^+,\what R_n^-)\Rightarrow (\zeta_{\alpha_+},C^{-1}\zeta_{\alpha_-})$. By the continuous mapping theorem, we have proved~\eqref{eq:kappanqn}.\ifthenelse{\boolean{format}}{\qed}{}
\end{proof}
Corollary~\ref{coro:power} provides theoretical support for the gamma test. Indeed, it shows that as long as $\kappa_n^\pm\sim(1/\gamma_\pm)^\leftarrow(n) = o(n)$, then $nq_n$ explodes quickly as $n\to\infty$. This means in this case, the gamma test rejects the null hypothesis with probability going to one as the sample size increases to infinity. Furthermore, the following example shows that when $\kappa_n^\pm\sim C_\pm n$ for some constants $C_\pm>0$, the gamma test still performs reasonably well. In effect, this is indeed the `worst' case that the gamma test could encounter, provided the ratio tail index exists.
\begin{Example}\label{example:power}
Recall the model~\eqref{eq:rho} considered in Example~\ref{example:patho}:
\[
(X,Y) \stackrel{\rm d}{=} ((1-\rho)Z_2\vee \rho Z_3,\rho Z_1\vee (1-\rho)Z_2)\,.
\]
In this case, $\gamma_\pm(t) = \rho/t$ for $t>1$, and the tail dependence index $\lambda = 1-\rho$. Corollary~\ref{coro:power} yields
\[
nq_{n}\Rightarrow \frac1{\rho}\bpp{\frac1{\zeta_1\topp1}+\frac1{\zeta_1\topp2}}\stackrel{\rm d}{=} \Gamma(2,\rho^{-1})\,.
\]
Now, observe that in the strong tail dependence case, i.e., $\rho$ is close to 0, the power of the test is high. Consider a test of level $0.05$. Let $q_\rho(\beta), \beta\in[0,1]$ denote the lower $\beta$ quantile of the distribution of $\Gamma(2,\rho^{-1})$. For this model, the power converges to
\begin{equation}\label{eq:limitPower}
1 - q^{-1}_\rho(q_1(0.95))\,.
\end{equation}
Figure~\ref{fig:power} illustrates the power of the test as a function of $\rho\in[0,1]$. We see that the test performs reasonably well as long as the tail dependence index $\lambda = 1-\rho$ is not too small. In addition, we also observe that the test statistic $nq_{n}$ converges quickly. For $n = 20$, the power of the test is already close to the limit one in~\eqref{eq:limitPower}.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width = 0.8\textwidth]{power.eps}
\end{center}
\caption{\label{fig:power}Power of the gamma test (solid line) in the `worst scenario' (Example~\ref{example:power}) for selected $\rho$'s with $n = 20$, based on 1000 simulations. The dashed line corresponds to the limit power calculated in~\eqref{eq:limitPower}.}
\end{figure}
\begin{Rem}
In this experiment, we used a {\em modified gamma test} instead of the original one. Namely, instead of $q_n = q_{0,n}$ defined as in~\eqref{eq:qun}, we used a slightly different statistic $\what q_{n} \mathrel{\mathop:}= R_n^+R_n^-/(R_n^+ + R_n^-)$ with $R_n^\pm$ as in~\eqref{eq:Rn}. We do not use the original test statistic $q_{n}$ because, in this example when $\rho$ is small and the sample size $n$ is small, with very high probability the $q_{0,n}$ would equal $0/0$, which is not well defined.

One can show that $nq_n$ and $n\what q_n$ have the same asymptotic distribution. Indeed, if we replace $q_n$ by $\what q_n$ in Corollary~\ref{coro:power}, all the statements remain valid. The proof will be the same, except that we have $\kappa_n^+\what q_n = ({\what R_n^++\what R_n^-})/(\what R_n^+\what R_n^-)$
instead of~\eqref{eq:kappan+qn}.
\end{Rem}
\end{Example}
\begin{Rem}\label{rem:power}
The model~\eqref{eq:rho} represents the `worst' case that the gamma test can encounter.
This is the case when the power of the gamma test is low, but the tail dependence is the strongest possible.

To design such a scenario, we need $nq_{n}$ to converge and $(X,Y)$ to have the largest tail dependence index possible. To have $nq_n$ to converge, by Corollary~\ref{coro:power} we must have $\kappa_n^\pm\sim C_\pm n$, which is equivalent to the fact that $H$ has point masses at $\{0\}$ and $\{1\}$. Now, the tail dependence index is maximized by concentrating all the measure of $H$ on $(0,1)$ at $\{1/2\}$.

In this example, the gamma test performs poorly when $\rho$ is close to 1. However, one should not expect any independent test to perform well, as $(X,Y)$ as in~\eqref{eq:rho} can be seen as a pair of independent random variables $(\rho Z_1,\rho Z_3)$, slightly perturbed by $(1-\rho)Z_2$ (via the `max'($\vee$) operation).

\end{Rem}
In \cite{zhang08quotient}, other versions of $q_n$ were also proposed, aiming at dealing with arbitrary bivariate distributions. In principle, the data needs to be first transformed to have 1-Fr\'echet marginals. It is an intriguing problem to characterize how the dependence structure changes during such a transformation.
The investigation along this line is out of the scope of this paper.
\begin{Rem}\label{rem:transformation}
Observe that for continuous random variables $X$ and $Y$ with c.d.f.~$F_X$ and $F_Y$, the joint distribution of $(-1/\log F_X(X), -1/\log F_Y(Y))$ will have marginal standard 1-Fr\'echet distributions. However, most of the time this transformation does not lead to a bivariate 1-Fr\'echet distribution.
For studies on these distributions, see e.g.~\cite{ledford96statistics,ledford97modelling}.
\end{Rem}

\section{Proof of Proposition~\ref{prop:1}}\label{sec:proofs}
We borrow a result from~\cite{weintraub91sample}. Weintraub's work is based on the min-stable distribution (processes), which can be equivalently transformed into our setting. Namely, if $(V,W)$ has a min-stable distribution with spectral functions $f$ and $g\in L^1(S,\mu)$ according to~\cite{weintraub91sample} (see also~\cite{dehaan86stationary}), then $(X,Y) \mathrel{\mathop:}= (1/V,1/W)$ is jointly 1-Fr\'echet with representation in~\eqref{eq:XY}.

Now, Lemma 3.4 in \cite{weintraub91sample} becomes, for $(X,Y)$ as in~\eqref{eq:XY},
\begin{eqnarray}
& & \mathbb P(X\leq x\mid Y = y) \nonumber\\
& & \ \ \ \  = \exp\bbb{-\int_S\bpp{\frac{f(s)}x-\frac{g(s)}y}{\bf 1}_{\ccbb{\frac{f(s)}{g(s)}> \frac xy}}\mu({\rm d} s)}\int_Sg(s){\bf 1}_{\ccbb{\frac{f(s)}{g(s)}\leq \frac xy}}\mu({\rm d} s)\nonumber\\
& & \ \ \ \  = \nn g_{D_t}\exp\bpp{-x^{-1}\nn f_{E_{x/y}}+y^{-1}\nn g_{E_{x/y}}}\,.\label{eq:weintraub}
\end{eqnarray}
Then, letting $\mathbb P_Y$ denote the distribution of $Y$,
\begin{eqnarray*}
\mathbb P(X/Y\leq t, Y>u) & = & \int_u^\infty\mathbb P(X\leq tr\mid Y = r)\mathbb P_Y({\rm d} r)\\
& = & \int_u^\infty \mathbb P(X\leq tr\mid Y = r)r^{-2}\exp(-r^{-1}){\rm d} r\\
& = & \nn g_{D_t}\int_u^\infty r^{-2}\exp\bpp{-\frac{1+t^{-1}\nn f_{E_t} - \nn g_{E_t}}r}{\rm d} r\,.
\end{eqnarray*}
Observe that $1- \nn g_{E_t} = \nn g_{D_t}$. We have thus obtained~\eqref{eq:prop1}. The proof is complete.
\begin{Rem}
Careful readers may find the difference between our definition of $D_t$ in~\eqref{eq:Dt} and the one in~\cite{weintraub91sample}, Lemma 3.4 (where different symbols are used). Therein, $D_t = \{s\in S:f(s)/g(s) < t\}$, with the `$\leq$' in~\eqref{eq:Dt} replaced by `$<$'. After reading the two paragraphs of the proof of Lemma 3.4, one should see that the correct definition of $D_t$ is as in~\eqref{eq:Dt} with `$\leq$'. See also the proof of Lemma 3.5 in~\cite{weintraub91sample}, where Lemma 3.4 was applied with `$\leq$'.

An alternative way to see quickly it is not correct to choose the definition with `$<$' is given next. Note that, using `$<$' instead of `$\leq$' would only lead to a different formula when $\nn g_{D_t}$ (or $\nn f_{E_t}$, resp.) is discontinuous at some $t_0>0$.
Consider
\[
F(t) \mathrel{\mathop:}= \mathbb P(X/Y\leq t) = \mathbb P(X/Y\leq t, Y>0) = \bpp{1+\frac {\nn f_{E_t}}{t\nn g_{D_t}}}^{-1}\,,
\]
which has jumps at $t_0$ such that $\mu\{s:f(s)/g(s) = t_0\}>0$. In this case, the definition with `$<$' would cause $F(t)$ to be right-discontinuous at $t_0$. Indeed, both $\nn f_{E_t}=\int_{E_t} f{\rm d}\mu$ and $\nn g_{D_t}=\int_{D_t} g{\rm d}\mu$ would be right-discontinuous at $t_0$. But $F(t)$, as a cumulative distribution function, should be right-continuous, which is a contradiction.

\end{Rem}
\section{Discussion}\label{sec:discussion}
Recently, several extensions of the notion of regular variation have been introduced. The main motivation behind them is to study in more details the asymptotic behaviors that are not captured by the standard multivariate regular variation.
In particular,~\cite{resnick02hidden} introduced the notion of hidden regular variation, characterizing the dependence structure when components of the random vectors are asymptotically independent but not independent;~\cite{maulik02asymptotic} investigated the regular variation of products of random variables, and applied the result to model the network traffic.
We discuss our results from these perspectives.
\medskip\\
{\it Hidden regular variation.}
Recall the definition of regular variation in the language of vague convergence~\eqref{eq:VC}. When the exponent measure $\nu$ concentrates on the two axes, the two random variables $V$ and $W$ are asymptotically independent. In this case,
$\nu$ does not provide useful information to characterize different dependence structures for asymptotically independent random variables. More sophisticated models are needed.

This problem was first investigated by \cite{ledford96statistics,ledford97modelling}, and their models later on were generalized under the framework of hidden regular variation by \cite{resnick02hidden,resnick07heavy,resnick08multivariate}. Hidden regular variation is present when the vague convergence~\eqref{eq:VC} holds with $\nu$ concentrated on the axes, i.e., $V$ and $W$ are asymptotically independent, and in addition there exists $\kappa_t^o = o(\kappa_t)$, such that
\[
t\mathbb P\bbb{\bpp{\frac V{\kappa_t^o},\frac W{\kappa_t^o}}\in\cdot} \stackrel{v}{\longrightarrow}\nu^o(\cdot)
\]
in $M_+({\mathbb E}_0)$ with ${\mathbb E}_0 = (0,\infty]\times (0,\infty]$. Intuitively, the notion of hidden regular variation involves normalizing the vector $(V,W)$ by sequences of constants of smaller order $(\kappa_t^o = o(\kappa_t)$) than required in~\eqref{eq:VC}. Thus, ignoring the two axes, certain dependence structure might appear in the limit. This dependence structure is not captured, therefore `hidden', when the `standard' rate $\kappa_t$ is taken.

In our case, when $(X,Y)$ is a bivariate 1-Fr\'echet random vector, we will see that $(V,W) = (X/Y,Y)$ are asymptotically independent, and there is no hidden regular variation. We first look at the convergence in form of
\begin{equation}\label{eq:X/YY}
\nu_t(\cdot)\mathrel{\mathop:}= t\mathbb P\bbb{\bpp{\frac{X/Y}{\kappa_t},\frac Yt}\in\cdot}\stackrel{v}{\longrightarrow}\nu(\cdot)
\end{equation}
in $M_+({\mathbb E})$ with $\kappa_t\sim(1/\gamma)^\leftarrow(n)$.
Here, we are in a slightly different situation, as the normalizing sequences $\kappa_t$ and $t$ are of different rates most of the time. In this case, we say $X/Y$ and $Y$ are asymptotically independent {\it with different rates}. The rates are the same, i.e., $\kappa_t\sim ct$ for some constant $c\in(0,\infty)$, if and only if $\int_Sf{\bf 1}_{\{g=0\}}{\rm d}\mu > 0$, by Corollary~\ref{coro:2}, part (i). Otherwise, $\kappa_t = o(t)$.

We first show below that $X/Y$ and $Y$ are asymptotically independent (with different rates if $(1/\gamma)^\leftarrow(t) = o(t)$). Indeed, for $r>0, y>0$, Proposition~\ref{prop:1} implies, writing $\wt t = \kappa_tr$ for simplicity,
\begin{eqnarray}
\nu_t([0,r]\times(y,\infty])& = & t\mathbb P\bpp{\frac{X/Y}{\kappa_t}\leq r,\frac Yt>y} \nonumber\\
& = & t\frac{\nn g_{D_{\wt t}}}{\nn g_{D_{\wt t}} + {\wt t}^{-1}\nn f_{E_{\wt t}}}\bbb{1-\exp\bpp{-\frac{\nn g_{D_{\wt t}} + \wt t^{-1}\nn f_{E_{\wt t}}}{ty}}}\nonumber\\
& = & \frac{\nn g_{D_{\wt t}}}y + o(1) = \frac 1y + o(1)\,,\mbox{ as } t\to\infty\,.\label{eq:t}
\end{eqnarray}
It follows that for all rectangles $E = [r_1,y_1)\times[r_2,y_2)$ in $\mathbb E_0$, $\lim_{t\to\infty}\nu_t(E) = 0$.
Recall also that by~\eqref{eq:convergence} (with $u=0$), $t\mathbb P(X/Y/\kappa_t>r)\sim r^{-\alpha}$. This together with~\eqref{eq:t} yields the following result.
\begin{Prop}
Let $(X,Y)$ be as in~\eqref{eq:XY} and suppose~\eqref{eq:RV} holds. Then, for $\kappa_t\sim(1/\gamma)^\leftarrow(t)$, the vague convergence~\eqref{eq:X/YY} holds in $M_+({\mathbb E})$ with $\nu$ concentrated on the two axes. In other words,
\[
\bpp{\frac1{\kappa_n}\bigvee_{i=1}^n\frac{X_i}{Y_i}, \frac1n \bigvee_{i=1}^n Y_i}\Rightarrow (\zeta_\alpha\topp1,\zeta_1\topp2)\,,
\]
where $\zeta_\alpha\topp1$ and $\zeta_1\topp2$ are independent standard Fr\'echet random variables with indices $\alpha$ and 1, respectively.
\end{Prop}
Now, we examine the hidden regular variation of this model. Suppose that there exist sequences $\kappa_t^o = o((1/\gamma)^\leftarrow(t))$ and $\iota_t^o = o(t)$, such that
\begin{equation}\label{eq:nuto}
\nu_t^o(\cdot)\mathrel{\mathop:}= t\mathbb P\bbb{\bpp{\frac{X/Y}{\kappa_t^o},\frac Y{\iota^o_t}}\in\cdot} \stackrel{v}{\rightarrow}\nu^o(\cdot)
\end{equation}
in $M_+(\mathbb E_0)$ for some non-zero $\nu^o\in M_+(\mathbb E_0)$, i.e., $(X/Y, Y)$ is hidden-regularly varying. Then, by a similar calculation as in~\eqref{eq:t}, we have
\[
\nu_t^o((0,r]\times(y,\infty]) = \frac t{\iota_t^oy}\nn g_{D_{\kappa_t^or}}(1+o(1)).
\]
We see that $\nu_t^o$ cannot converge in $\mathbb E_0$. Therefore, we have proved the following.
\begin{Prop}
There is no hidden regular variation for $(X/Y, Y)$.
\end{Prop}
\noindent{\it Regular variation of products.}
\cite{maulik02asymptotic} proposed different generalizations of~\eqref{eq:VC}, for the purpose of characterizing the regular variation of the product of random variables. Their main results focus on two different situations.

First, if $V\in RV_{-\alpha}$ and
\begin{equation}\label{eq:AI}
t\mathbb P\bbb{\bpp{\frac V{\kappa_t},W}\in\cdot}\stackrel{v}{\longrightarrow}(\nu_\alpha\times G)(\cdot)
\end{equation}
in $M_+(\mathbb D), \mathbb D = (0,\infty]\times[0,\infty]$, where $\nu_\alpha((x,\infty]) = x^{-\alpha}, x>0$ and $G$ is a probability measure with $G(0,\infty) = 1$. Then,
\[
t\mathbb P\bbb{\frac{(V,VW)}{\kappa_t}\in\cdot}\stackrel{v}{\longrightarrow}\nu(\cdot)
\]
in $M_+(\mathbb D)$ with $\nu\in M_+(\mathbb D)$, determined by $\nu_\alpha$ and $G$ (\cite{maulik02asymptotic}, Theorem 2.1). This case can be seen as a generalization of Breiman's theorem, since $VW$ has an equivalent tail as $V$. Indeed,~\eqref{eq:AI} is referred to as the (new) definition of asymptotic independence of $V$ and $W$ therein.

Second, suppose $V\in RV_{-\alpha_V}, W\in RV_{-\alpha_W}$ for some $\alpha_V,\alpha_W>0$ and $V$ and $W$ are asymptotically {\it dependent} with different rates, or equivalently,
\[
t\mathbb P\bbb{\bpp{\frac V{\kappa_t},\frac W{\iota_t}}\in\cdot} \stackrel{v}{\longrightarrow} \nu(\cdot)
\]
in $M_+({\mathbb E}_0)$ with $\nu((0,\infty]^2) > 0$. Then, $VW\in RV_{-({\alpha_V\alpha_W})/({\alpha_V+\alpha_W})}$.
In this situation, the multiplication with a random variable changes the tail behavior.

Our asymptotic result on the bivariate Fr\'echet random variables (Theorem~\ref{thm:RV}, with $V = X$ and $W = 1/Y$), does not fall into any of these situations. Indeed, if~\eqref{eq:RV} holds with $\alpha>1$, then $X/Y$ has a lighter tail than $X$, which differs from the first situation; on the other hand, $1/Y$ always has exponential distribution, which is not regularly varying, whence we are not in the second situation either. Therefore, the following question arises.
\begin{Ques}\label{ques:1}
Let $V$ and $W$ be two nonnegative random variables. Suppose $V\in RV_{-\alpha}$ (but $W$ may not be regularly varying). Provide a sufficient condition on $V$ and $W$ such that $VW$ is regularly varying with tail index $\beta>\alpha>0$.
\end{Ques}
\ifthenelse{\boolean{format}}{
\begin{acknowledgements}
The author was grateful to Stilian Stoev for his careful reading of an early version of the paper, as well as many inspiring and helpful comments and suggestions. The author would also like to thank the Editor J\"urg H\"usler and two anonymous referees for their helpful suggestions and comments, which significantly improved the paper.
\end{acknowledgements}}
{\vspace{0.2in}
\noindent{\bf Acknowledgements}
The author was grateful to Stilian Stoev for his careful reading of an early version of the paper, as well as many inspiring and helpful comments and suggestions. The author would also like to thank the Editor J\"urg H\"usler and two anonymous referees for their helpful suggestions and comments, which significantly improved the paper. The author was partially supported by the National Science Foundation grant DMS--0806094 at the University of Michigan.
}
\def\doi#1{DOI: #1}

\def$'${$'$}

redpajama_set_name: RedPajamaArXiv


text:
> Mental Health Service Wins National PIP Award
Mental Health Service Wins National PIP Award
A carers' forum which provides support to families and friends of patients at Rowanbank Clinic and Leverndale Hospital has won a prestigious national award.
The Principles into Practice Awards (PiP) celebrate the best practice in mental health services and the Charing Cross Forensic Carers Group have been judged best in the Carer Involvement and Support Award.
The group is facilitated by Caroline Peddie, Lead Carers Facilitator, Rowanbank Clinic, and Maryanne Wilkie, Deputy Carer Facilitator. Caroline is delighted the group has been recognised for the significant peer support they have provided for the last three years.
She said: "Winning this award is wonderful for the group who have worked tirelessly over the last three years supporting families and friends of patients within the Forensic Directorate.
"Often family members and friends feel alone when they have a loved one as an inpatient in a mental health facility. What the group does is offer people a chance to share their experiences with other people in a similar position."
One of the carers involved with the group, Christine, added: "I have leaned a lot from the group, but the most important thing of all, is to know that you are not alone…"
The Principles into Practice Network sets out how professionals should work with patients and carers. For professionals, the principles provide a framework to help patients consider the ethical aspects or alternative approaches to their care and treatment.
For further information either telephone 0141 201 4429 or email [email protected] .
Find A&E and MIU services by postcode

redpajama_set_name: RedPajamaCommonCrawl


0:08 Substitution!Findlay substitution: Anna Money for Bre Parrish.
0:22 Anna Money at goalie for Findlay.
05:43 Offside against Kentucky Wesleyan.
07:40 Shot by Kentucky Wesleyan HUNT, MADISYN, Save (by goalie) Anna Money.
08:41 Corner kick by Findlay Samantha Church [08:41].
09:08 Goal!GOAL by Findlay Ashley Rings (FIRST GOAL), Assist by Samantha Church and Lindsey Ahbe, goal number 2 for season. *goal scored from corner kick.
10:53 Corner kick by Findlay Samantha Church [10:53].
11:38 Shot by Findlay Ashley Rings, Wide Right.
11:54 Substitution!Kentucky Wesleyan substitution: BURNETTE, KALEY for HUMPHRIES, KELSEY.
12:17 Substitution!Kentucky Wesleyan substitution: O'NAN, DARBY for GORDON, ELIZABETH.
13:57 Shot by Findlay Lindsey Roeth, Wide Right.
16:35 Shot by Findlay Samantha Church, Wide Right.
21:00 Corner kick by Kentucky Wesleyan HAASE, BREIGH [21:00].
23:57 Goal!GOAL by Kentucky Wesleyan HAASE, BREIGH, Assist by HAYCRAFT, MADISON *one-time goal to left side of net.
26:57 Corner kick by Kentucky Wesleyan HAASE, BREIGH [26:57].
27:30 Shot by Kentucky Wesleyan HAASE, BREIGH, Save (by goalie) Anna Money.
28:57 Substitution!Findlay substitution: Karleigh Clere for Samantha Church.
29:54 Shot by Findlay Sierra Peacock, Wide Right.
31:36 Shot by Kentucky Wesleyan HUNT, MADISYN, Save (by goalie) Anna Money.
34:31 Shot by Findlay Megan Kupetz, Save (by goalie) RENTCHLER, TAYLOR.
36:30 Shot by Kentucky Wesleyan HUNT, MADISYN, High.
38:22 Shot by Kentucky Wesleyan HAASE, BREIGH, High.
40:12 Shot by Findlay Ashley Rings, Wide Right.
40:55 Offside against Kentucky Wesleyan.
43:41 Header Shot by Findlay Karleigh Clere, High.
43:47 Substitution!Kentucky Wesleyan substitution: BOONE, ALLY for HUNT, MADISYN.
45:00 For Kentucky Wesleyan: #25 HUNT, MADISYN, #14 O'NAN, DARBY, #15 WALDEN, SKYLAR, #21 DUFF, ANNA, #10 PRUITT, SIDNEY, #1 RENTCHLER, TAYLOR, #8 ZETTER, ERICA, #5 HAYCRAFT, MADISON, #20 HAASE, BREIGH, #22 HINCHCLIFFE, ERIKA, #7 BURNETTE, KALEY.
45:00 For Findlay: #0 Anna Money, #19 Ashley Rings, #9 Sierra Peacock, #17 Samantha Church, #3 Megan Kupetz, #12 Lindsey Ahbe, #4 Ashley Zenker, #15 Karis Dewalt, #6 Lindsey Roeth, #10 Karleigh Clere, #5 Madison VanCucha.
50:36 Shot by Findlay Ashley Rings, Save (by goalie) RENTCHLER, TAYLOR.
50:47 Corner kick by Findlay Samantha Church [50:47].
51:25 Shot by Kentucky Wesleyan HUNT, MADISYN, Save (by goalie) Anna Money.
56:15 Header Shot by Findlay Karleigh Clere, Wide Right.
56:50 Shot by Findlay Sierra Peacock, High.
57:58 Shot by Kentucky Wesleyan HAASE, BREIGH, Save (by goalie) Anna Money.
59:10 Shot by Findlay Ashley Rings, Wide Right.
59:22 Substitution!Findlay substitution: Jensen Kleis for Sierra Peacock.
59:22 Substitution!Kentucky Wesleyan substitution: GORDON, ELIZABETH for O'NAN, DARBY.
59:22 Substitution!Kentucky Wesleyan substitution: HUMPHRIES, KELSEY for HAYCRAFT, MADISON.
61:01 Shot by Findlay Jensen Kleis, Save (by goalie) RENTCHLER, TAYLOR.
64:33 Shot by Findlay Jensen Kleis, Save (by goalie) RENTCHLER, TAYLOR.
66:22 Substitution!Kentucky Wesleyan substitution: HAYCRAFT, MADISON for BURNETTE, KALEY.
66:22 Substitution!Findlay substitution: Sierra Peacock for Ashley Rings.
67:20 Substitution!Kentucky Wesleyan substitution: O'NAN, DARBY for GORDON, ELIZABETH.
69:01 Shot by Kentucky Wesleyan HUNT, MADISYN, Save (by goalie) Anna Money.
69:58 Shot by Findlay Jensen Kleis, Save (by goalie) RENTCHLER, TAYLOR.
70:42 Corner kick by Findlay Samantha Church [70:42].
71:36 Offside against Kentucky Wesleyan.
72:50 Shot by Findlay Sierra Peacock, Save (by goalie) RENTCHLER, TAYLOR.
74:20 Shot by Findlay Madison VanCucha, Save (by goalie) RENTCHLER, TAYLOR.
75:04 Corner kick by Findlay Samantha Church [75:04].
75:39 Shot by Kentucky Wesleyan HAASE, BREIGH, Wide.
75:49 Substitution!Kentucky Wesleyan substitution: BURNETTE, KALEY for HUMPHRIES, KELSEY.
75:49 Substitution!Kentucky Wesleyan substitution: HARRIS, TAYNA for O'NAN, DARBY.
77:43 Shot by Kentucky Wesleyan HAYCRAFT, MADISON, High.
83:03 Shot by Findlay Jensen Kleis, High.
83:39 Shot by Findlay Madison VanCucha, Hit Crossbar.
83:39 Goal!GOAL by Findlay Jensen Kleis, goal number 1 for season. *goal coming off of deflection of cross bar.
84:47 Substitution!Findlay substitution: Ashley Rings for Samantha Church.
89:52 Shot by Kentucky Wesleyan HINCHCLIFFE, ERIKA, Save (by goalie) Anna Money.


text:
Language learning apps
Find all the best apps to learn any language you would like to speak fluently! Free download for iOS and Android devices from iTunes & Google Play.
12 Best apps to learn French for iOS & Android
Learning languages today is easier with numerous apps. Some of them have special courses, other suggest learn language through games. But if you are interested ...
20 Best apps to learn Japanese for Android & iOS
With the proliferation of smartphones and the simplification of access to the Internet, there is a large number of applications that help in learning different ...
15 Best Apps to Learn Spanish for Android & iOS
Many people have difficulty with self-studying foreign languages. Often, we just do not have the concentration and perseverance to study the material ...
Fresh apps listings:
Dark Fear App Review
11 Best Monster Catching Games for Android & iOS
15 Best Point and Click Adventure Games for Android & iOS
15 Free Anime Mobile Games for Android & iOS 2021
11 Free Multiplayer Puzzle Games for Android & iOS
Free apps reviews
© Free apps for iOS & Android. 2016-2021 If using any material from this website - dofollow hyperlink required.


For external or internal use. The T5000 Marine is designed to attain a high level of noise reduction (up to 23dB). The hood is manufactured using general-purpose resin material to BS476 Class 1 fire regulation.
The T5000 Marine is extremely robust and weatherproof. Inside the hood's outer shell is a perforated polypropylene lining that houses 50mm non-hygroscopic acoustic insulation.
Fitted to the inner back wall is an apparatus plate; beneath this plate is a detachable GRP shelf (165mm deep) with easy cable entry. The hood is fixed to the wall via two brackets that are secured through the hood's back panel enabling a load of up to 60 kgs to be safely mounted.
Designed for use in noisy areas and harsh environments such as off shore oil rigs and newspaper print rooms.
Inside the hoods outer shell is a perforated polypropylene lining that houses 50mm non- hygroscopic acoustic insulation.
High visibility yellow paint finish…other colours can be made to order.
Price now includes grade 316 stainless steel mounting brackets and apparatus plate.
Postage – Please choose the postage "Acoustic Hood – (T4000, T5000, T5000 Marine, GRP Kiosk)" option at the checkout. This product is supplied on a pallet and is very heavy.


Cumbria Victims Charitable Trust provides support to Victims of crime who reside in Cumbria.
Who can apply? Any victim of crime who resides in Cumbria and has been affected by crime or antisocial behaviour.
What the fund will support?
What the fund will NOT support?
Grants will usually be around £500 however there is no minimum or maximum award.


text:
Fatal Frame: Maiden of Black Water Launches Digitally on October 28th
One of the playable characters, Yuri Kozukata, using The Camera Obscura.
There has been a lot of anticipation at the return of Fatal Frame. A classic in the survival horror genre, it has been exclusive to Nintendo since 2008 with the Wii exclusive Mask of the Lunar Eclipse. So fans and myself were excited that when they announced a Remake of the WiiU exclusive Fatal Frame: Maiden of Black Water. And Koei Tecmo has confirmed that Fatal Frame: Maiden of Black Water launches digitally on October 28th.
Alongside a release date, we got more information on a Deluxe Edition of the game. The Digital Deluxe edition will include a copy of the game, a digital artbook, and a classic costume set. The classic costume set is inspired by the protagonists of previous entries in the series.
And for users that purchase the game within its first two weeks of launch; will receive a special Ryza costume based on Atelier Ryza: Ever Darkness & the Secret Hideout. Along with the announcement came a new 4-minute overview trailer. This provides some information to the setup for Maiden of Black Water seen below:
Where and When is it coming?
While being an enhanced version of the much overlooked WiiU version, Fatal Frame: Maiden of Black Water, will also be a multiplatform release. This is the first time that Fatal Frame has been on non-Nintendo consoles since the third installment. Maiden of Black Water launches on October 28th on Xbox One, Xbox Series X/S, Playstation 4/5, Switch, and Steam. For more information, you can find it on the official Fatal Frame website.
-Jess (Bloodieknux)
Hey, guys hope you enjoy my thoughts and breakdowns in the articles above. If you wish to keep up with all my shenanigans and my live streams you can follow me on all my social media listed: Twitter, Facebook, Instagram, Twitch, & Glimesh.
On July 30, 2021 By Jesus Cisnero
Previous Post10 Million PlayStation 5 Sold | Fastest Selling Xbox | Pax West Restrictions – 2XP LVL 165
Next PostMortal Shell Gets A New DLC 'The Virtuous Cycle'


Location: Saint-Étienne-Vallée-Française, in the Lozere departement of France (Languedoc-Roussillon Region).
Notes: Château de Cambiaire was built in the fourteenth century in the town of Saint-Étienne-Vallée-French, department of Lozere. This castle stands just above the village of Saint-Étienne-Vallée-French Lozère. It is a quadrangular building with a round tower at three of its corners and a square tower at the northwest corner. It consists of three wings around a courtyard on the west side by a battlements surrounding wall pierced by a gate. The great crenellated tower, which dungeon office, is crowned on its summit terrace of a watchtower with a roof shaped pepper shakers. This square building built shale stone mixed with white quartz is the oldest. It has three levels, the first two date from the fourteenth century, the last level and the spiral staircase from a raising of the late fifteenth century. This tower seems more ostentatious than most military as much as the castle was never the castle of Saint-Etienne. One of the ground pavement dwelling is probably contemporary dungeon. The others are the result of rearrangements of the modern era and those of the late nineteenth century and finally repairs arising out the 1944 fire. A park of chestnut trees and cedars surrounding it. There is a replica of the Lourdes grotto and a chapel dating back to 1875 in neo-Gothic style.