# On the relation between extremal dependence and concomitants

Amir Khorrami<EMAIL_ADDRESS>[ Marie
<EMAIL_ADDRESS>[ ESOMAS Department, University of Torino,
Torino, Italypresep=, ]e1  ESSEC Business School, CREAR, Francepresep=, ]e2

###### Abstract

The study of concomitants has recently met a renewed interest due to its
applications in selection procedures. For instance, concomitants are used in
ranked-set sampling, to achieve efficiency and reduce cost when compared to
the simple random sampling. In parallel, the search for new methods to provide
a rich description of extremal dependence among multiple time series has
rapidly grown, due also to its numerous practical implications and the lack of
suitable models to assess it. Here, our aim is to investigate extremal
dependence when choosing the concomitants approach. In this study, we show how
the extremal dependence of a vector $(X,Y)$ impacts the asymptotic behavior of
the maxima over subsets of concomitants. Furthermore, discussing the various
conditions and results, we investigate how transformations of the marginal
distributions of $X$ and $Y$ influence the degeneracy of the limit.

60G70,

62E20,

62G30,

Asymptotic Theorems,

Concomitants,

Copula,

(Tail) Dependence,

Extremes,

Gaussian,

Logistic,

Maxima,

Order Statistics,

Pareto,

Slowly/Regularly Varying Functions,

Tail Equivalence,

Weak Convergence,

###### keywords:

[class=MSC]

###### keywords:

,

##  1 Introduction

Let $(X_{i},Y_{i}),\,i=1,\dots,n$, be a sequence of independent and
identically distributed (i.i.d.) bivariate random vectors with common
cumulative distribution (cdf) function $F(x,y)$. We define the _concomitants
of order statistics_ as follows. Arrange the $X$ variates in ascending order,
thus obtaining the order statistics $X_{(1)}\leq X_{(2)}\leq\dots\leq X_{(n)}$
for the $X$ variable. Then, the $Y$ variable corresponding to the $i$-th order
statistic $X_{(i)}$ is called the concomitant of the $i$-th order statistic,
and is denoted by $Y_{[i]}$. For example, if the maximum of the $X$ variates
is $X_{(n)}\equiv X_{3}$, then its concomitant is $Y_{[n]}\equiv Y_{3}$.
Concomitants of order statistics are of notable interest in practical
applications: It is worth mentioning their utility in problems concerning the
estimation of parameters for multivariate datasets affected by type II
censoring, in the development of selection procedures (where it is more
convenient to measure a characteristic linked to a measure of interest), in
the setting of ranked-set sampling schemes. In particular, the latter deals
with situations where the measurement of the variable of interest is time-
consuming or expensive, but the ranking of a set of observations linked to the
target variable is easy to be performed. For such cases, ranked-set sampling
is a better alternative with respect to the simple random sampling, both in
terms of efficiency and reduced cost, hence this renewed attention on the
study of concomitants. We report the seminal works Dell and Clutter (1972),
Stokes and Sager (1988), and more recently Wang et al. (2006), Shao (2010),
Balci, Akkaya and Ulgen (2013), Wang, Lim and Stokes (2016), Wang, Ahn and Lim
(2017), Zamanzade and Wang (2018), Ozturk (2019) among others.

Our motivation is to investigate extremal dependence when choosing the
concomitant approach. Looking at the literature, this topic has been tackled
in the 90’s. One can mention the seminal paper Nagaraja and David (1994)
where, given a number $k$, they study the distribution of the maximum of the
concomitants $V_{1}=\max\left(Y_{[n]},\dots,Y_{[n-k+1]}\right)$. Finite and
asymptotic results are given in both cases $k$ a fixed integer (_extreme
case_) and $k=[np],\,p\in(0,1)$ (_quantile case_). This leads to the natural
question of how much $V_{1}$ is close to the maximum $Y_{(n)}$, a fundamental
problem in practical applications, tackled in Joshi and Nagaraja (1995). In
their work, authors propose to link this question to the study of the joint
distribution of two maxima of concomitants. More precisely, given an i.i.d.
sequence of bivariate random vectors with parent random vector $(X,Y)$,
define, for $k\geq 1$,

$V_{1}=\max\left(Y_{[n]},\dots,Y_{[n-k+1]}\right)\quad\text{and}\quad
V_{2}=\max\left(Y_{[n-k]},\dots,Y_{[1]}\right).$ (1)

Joshi and Nagaraja (1995) introduced the random variable
$W_{k}:=V_{1}/Y_{(n)}$ and proved that its cdf satisfies

$\mathbb{P}\\!\left(W_{k}\leq w\right)=\mathbb{P}\\!\left(V_{1}\leq
wV_{2},\,V_{2}>0\right),$

for which they need to compute the joint distribution of $(V_{1},V_{2})$.

Before briefly reporting their result, let us introduce some notations that
will be used throughout the paper.
Let $G(z;\mu,\sigma,\xi)$, for $\mu,\xi\in\mathbb{R},\sigma>0$, denote a
Generalised Extreme Value distribution (GEV). A distribution function $F$ is
said to belong to the _Maximum Domain of Attraction_ (MDA) of $G$, written as
$F\in\mathscr{D}(G)$, if there exist normalizing sequences of constants
$a_{n}>0$ and $b_{n}\in\mathbb{R}$ such that $F^{n}(a_{n}z+b_{n})\to G(z)$, as
$n\to\infty$ for any $z\in\Gamma_{G}=\\{z\colon\xi(z-\mu)/\sigma>0\\}$. The
structure of a MDA is studied in Fisher and Tippett (1928), Gnedenko (1943),
among others, and more recently, with another view, in Leonetti and Khorrami
Chokami (2022). Sufficient conditions for a distribution $F$ to belong to the
MDA of a certain GEV $G$ are the so called Von Mises conditions: We refer to
(Resnick, 1987, Propositions 1.15 - 1.17) for a thorough description and
provide a brief recall in Appendix C for the paper to be self-contained.
A measurable function $h\colon\mathbb{R}_{+}\mapsto\mathbb{R}_{+}$ is
regularly varying at infinity with index $a$ (written $h\in\text{RV}_{a}$) if
for $x>0$

$\lim_{t\to\infty}\frac{h(tx)}{h(t)}=x^{a}.$

The index $a$ is called the _exponent of variation_. When $a$ is $0$, $h$ is
said _slowly varying_ (SV).
The _upper-tail dependence coefficient_ of a random vector $(X,Y)$ with copula
$C$ and marginal distributions $F_{X}$ and $F_{Y}$, respectively, is defined
as (Coles, Heffernan and Tawn (1999))

$\lambda_{u}:=\lim_{\alpha\to 1}\mathbb{P}\\!\left(Y>F_{Y}^{-1}(\alpha)\mid
X>F_{X}^{-1}(\alpha)\right)=\lim_{\alpha\to
1}\frac{1-2\alpha-C(\alpha,\alpha)}{1-\alpha},$ (2)

$F^{-1}$ denoting the generalized inverse function of a cdf $F$.
Note that $\lambda_{u}\in[0,1]$, where $\lambda_{u}=0$ means extremal
independence between $X$ and $Y$.
Let us also introduce the following notation, borrowed from Joshi and Nagaraja
(1995). Given a vector $(X,Y)$, we define

$\displaystyle F_{1}(y\mid x):=\mathbb{P}\\!\left(Y\leq y\mid X>x\right),$ (3)
$\displaystyle F_{2}(y\mid x):=\mathbb{P}\\!\left(Y\leq y\mid X\leq x\right),$
(4) $\displaystyle F_{3}(y\mid x):=\mathbb{P}\\!\left(Y\leq y\mid X=x\right).$
(5)

The conditional distributions $F_{1},\,F_{2}$ and $F_{3}$ play a fundamental
role in computing the distributions of the concomitants of the order
statistics as well as the distributions of their transformations.

Theorem 2 in Joshi and Nagaraja (1995): Suppose $F_{X}$ satisfies one of the
Von Mises conditions and, for all $x$ and $y$, assume
$F_{X}\in\mathscr{D}(G_{X})$, $F_{Y}\in\mathscr{D}(G_{Y})$, and

$n\mathbb{P}\\!\left(X>a_{n}x+b_{n},Y>c_{n}y+d_{n}\right)\underset{n\to\infty}{\to}0,\quad\forall
x,y\in\mathbb{R},$ (6)

where the norming constants
${(a_{n})}_{n},\,{(b_{n})}_{n},\,{(c_{n})}_{n}\text{ and }{(d_{n})}_{n}$ are
such that

$F_{3}(c_{n}y+d_{n}\mid a_{n}x+b_{n})\to 1\qquad\text{ as }n\to\infty.$ (7)

Further, suppose there exist constants $\tilde{a}_{n}>0$ and
$\tilde{b}_{n}\in\mathbb{R}$ such that

$F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid
a_{n}x+b_{n})\underset{n\to\infty}{\to}H_{1}(x,y).$ (8)

Then, the cdf of $(V_{1},V_{2})$ satisfies

$F_{(V_{1},V_{2})}(\tilde{a}_{n}v_{1}+\tilde{b}_{n},\,c_{n}v_{2}+d_{n})\underset{n\to\infty}{\to}H(v_{1})G_{Y}(v_{2}),\quad\forall
v_{1},v_{2}\in\mathbb{R},$ (9)

where $H(v_{1})={(k!)}^{-1}\int H_{1}^{k}(x,v_{1}){\left(-\log
G_{X}(x)\right)}^{k}g_{X}(x)\,\textrm{d}{x}$.

We begin our analysis by focusing on Condition (7). We first show in Lemma 1
the relation between (6) and (7), which proof is given in Appendix A.

######  Lemma 1.

Let $x,y\in\mathbb{R}$. Then, we have

$\displaystyle
n\mathbb{P}\\!\left(X>a_{n}x+b_{n},Y>c_{n}y+d_{n}\right)\underset{n\to\infty}{\to}0$
$\displaystyle\Leftrightarrow\quad F_{3}\left(c_{n}y+d_{n}\mid
a_{n}z+b_{n}\right)\underset{n\to\infty}{\to}1,\;\forall z\geq x.$ (10)

Now, we can see that (7) can be directly understood in terms of the upper-tail
dependence coefficient, namely:

######  Proposition 1.

Consider a bivariate random vector $(X,Y)$ with copula $C$. Then,

$F_{3}\left(c_{n}y+d_{n}\mid
a_{n}x+b_{n}\right)\underset{n\to\infty}{\to}1,\quad\forall x\geq
y\quad\Leftrightarrow\quad\lambda_{u}=0$

where the normalizing constants are chosen depending on the MDA to which $X$
and $Y$ belong.

Note that Proposition 1 is then simply another way to express Proposition 5.27
in (Resnick, 1987, p. 296), which states that

$\lambda_{u}=0\quad\Leftrightarrow\quad
n\mathbb{P}\\!\left(X>a_{n}x+b_{n},Y>c_{n}y+d_{n}\right)\underset{n\to\infty}{\to}0.$

Using Lemma 1 and Proposition 1, we can now rewrite Theorem 2 in Joshi and
Nagaraja (1995) in terms of $\lambda_{u}$. This reformulation, given in
Corollary 1, is helpful since it clarifies the connection between the
asymptotic independence of $(V_{1},V_{2})$ and that of $(X,Y)$.

######  Corollary 1.

Suppose $F_{X}$ satisfies one of the Von Mises conditions and, for all
$x,\,y$, $F_{X}\in\mathscr{D}(G_{X})$ and $F_{Y}\in\mathscr{D}(G_{Y})$.
Moreover, assume that the upper tail dependence coefficient between $X$ and
$Y$ is $0$: $\lambda_{u}=0$. If there exist constants
$\tilde{a}_{n}>0,\,\tilde{b}_{n}$ s.t. $F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid
a_{n}x+b_{n})\underset{n\to\infty}{\to}H(x,y)$, then the cdf
$F_{(V_{1},V_{2})}$ of $(V_{1},V_{2})$ satisfies (9).

The assumption of asymptotic independence between $X$ and $Y$ is generally too
restrictive in practical applications, especially in this highly
interconnected world. For instance, over the past 15 years, we have had to
live through a major global financial crisis, then a pandemic, combined with
an increase in cyber attacks. The presence of systemic risk, with a strong
dependence among extremes, is one characteristic of all those events (_e.g._
Dacorogna and Kratz (2015)). Hence, investigating this (asymptotic) dependence
aspect is of primary interest. This is the main objective of the next section.

##  2 Joint asymptotics for maxima over subsets of concomitants

We aim at studying the limiting distribution of the concomitants
$(V_{1},V_{2})$ when $X$ and $Y$ may exhibit some asymptotic dependence. To do
so, we take advantage of the joint-tail model representation given in Ledford
and Tawn (1998), then we state the main result (in Subsection 2.2) and discuss
the choice of the normalizing constants to avoid degeneracy in the limit.

###  2.1 The LT joint-tail model

First, let us recall some definitions related to the slowly varying notion in
the bivariate case. It will be needed to define the joint-tail model.

######  Definition 1.

A function $\mathcal{L}$ that satisfies

$\lim_{n\to\infty}\frac{\mathcal{L}(nx,ny)}{\mathcal{L}(n,n)}=r(x,y),\qquad\text{with
}r(ax,ay)=r(x,y),$ (11)

for all $a>0$ and $(x,y)\in\mathbb{R}^{2}_{+}$, is said to be _bivariate
slowly varying_ (BSV).

It can be shown that there exists a univariate function $r_{*}$ such that

$r(x,y)=r_{*}(w),\quad\mbox{with}\quad w=\frac{x}{x+y}.$ (12)

######  Definition 2.

A BSV function $\mathcal{L}$ for which the function $r_{*}$ defined in (12)
satisfies

$\frac{r_{*}(w)}{r_{*}(1-w)}\text{ is SV at $w=0$ and $w=1$},$

is said to be _quasi-symmetric_.

Let us now present the joint-tail model, named the LT joint-tail model,
introduced in Ledford and Tawn (1998) to study the asymptotic distribution of
the concomitant of the max, $Y_{[n]}$, i.e. the limit distribution of the
marginal distribution of $V_{1}$ (case $k=1$), and to compute
$\mathbb{P}\\!\left(Y_{(n)}=Y_{[n]}\right)$.

Let $(X,Y)$ be a bivariate random vector with unit Fréchet marginals (_i.e._
$F_{X}(x)=\textrm{e}^{-1/x}$, $x>0$) and joint distribution function $F(x,y)$.
Suppose that there exist functions $c(t)$ and $\psi(x,y)$ for which the joint
survival function

$\bar{F}(x,y)=1-\textrm{e}^{-1/x}-\textrm{e}^{-1/y}+F(x,y)$

satisfies

$\lim_{t\rightarrow\infty}\frac{t\bar{F}(tx,ty)}{c(t)}=\psi(x,y)\text{ for all
}(x,y)\in\mathbb{R}_{+}^{2}.$ (13)

Additionally suppose that there exists $\gamma\in\mathbb{R}$ such that

$\begin{cases}\bar{F}(0,t)/\bar{F}(t,0)&\text{ is }\text{RV}_{\gamma}\text{ at
infinity and}\\\ \bar{F}(t,0)/\bar{F}(0,t)&\text{ is
}\text{RV}_{-\gamma}\text{ at infinity}.\end{cases}$ (14)

Then the following holds.

######  Definition 3 (The LT joint-tail model).

Assume $(X,Y)$ have unit Fréchet marginals. Let $\eta\in(0,1]$ be the
_coefficient of tail dependence_ used to determine the decay rate of
$F^{-1}(t,t)$, as $t\to\infty$. Under Assumptions (13) and (14), the joint
survival distribution of $(X,Y)$ is given by

$\mathbb{P}\\!\left(X>x,Y>y\right)=\mathcal{L}(x,y)x^{-\alpha}y^{-\beta},\quad\forall
x,y,$ (15)

where $\alpha,\beta>0,\,\alpha+\beta=\eta^{-1}$ and the function
$\mathcal{L}(x,y)$ is a quasi-symmetric BSV function (see Definition 2).

The coefficient $\eta$ describes the type of limiting dependence between $X$
and $Y$, while the function $\mathcal{L}$ its relative strength given a
particular value of $\eta$; see Ledford and Tawn (1998) and Heffernan (2000)
for further comments. As pointed out in de Haan and Zhou (2011), the
$\eta=1/2$ corresponds to the case where $X$ and $Y$ are independent, while
the case $\eta>1/2$ indicates a positive association of the extremes of
$(X,Y)$. Values of $\eta$ lower than $1/2$ represent cases where the extremes
of $X$ and $Y$ are negatively associated. The bounding cases of perfect
negative and positive dependence correspond respectively to $\eta\to 0$ and
$\eta=1$ with $\mathcal{L}(\cdot)=1$. Coles, Heffernan and Tawn (1999)
introduce an elementary measure of dependence: Since the survival copula
$\bar{C}$ is given by $\bar{C}(u,v)=1-u-v+C(u,v)$, they consider the following
indices to describe the upper-tail dependence:

$\displaystyle\bar{\chi}:=\lim_{u\to
1}\frac{2\log(1-u)}{\log\bar{C}(u,u)}-1=2\eta-1$ (16)
$\displaystyle\lambda_{u}:=2-\lim_{u\to 1}\frac{\log C(u,u)}{\log
u}=\begin{cases}c&\text{ if }\bar{\chi}=1,\,\mathcal{L}(t)\to c>0,\,\text{as
}t\to\infty,\\\ 0&\text{ if }\bar{\chi}=1,\,\mathcal{L}(t)\to 0,\,\text{as
}t\to\infty,\\\ 0&\text{ if }\bar{\chi}<1.\end{cases}$ (17)

Since $\bar{\chi}=1$ if and only if $\eta=1$, we have a link between
$\lambda_{u}$ and $\eta$ as well. In other words,

  * •

$X$ and $Y$ are asymptotically independent when $\alpha+\beta=1$ and
$\mathcal{L}(n,n)\underset{n\to\infty}{\to}0$, or when $\alpha+\beta\geq 1$
(with no condition on the limit as $n\to\infty$ of $\mathcal{L}(n,n)$);

  * •

$X$ and $Y$ are asymptotically dependent when $\alpha+\beta=1$ and
$\mathcal{L}(n,n)\underset{n\to\infty}{\to}c>0$. Note that, in this case,
$c=\lambda_{u}$.

This turns out to be useful when considering extensions of Theorem 2 in Joshi
and Nagaraja (1995), relaxing Condition (7).

For the sake of simplicity, we will use the notation
$C_{\not\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}$ to indicate the asymptotic dependence case,
that is $\alpha+\beta=1$ and
$\mathcal{L}(n,n)\underset{n\to\infty}{\to}\lambda_{u}>0$. Thus, the indicator
$\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}=1$ points out the presence of asymptotic
dependence.

###  2.2 Main result

Let us state our main result on the joint asymptotic behaviour of maxima of
concomitants, allowing for asymptotic dependence of $X$ and $Y$:

######  Theorem 1.

Let $(X,Y)$ be a bivariate random vector with unit Fréchet marginals and
upper-tail dependence coefficient $\lambda_{u}$. Assume that $(X,Y)$ follows
the LT joint-tail model given in Definition 3, with survival cdf (15), and
that the BSV function $\mathcal{L}$ is such that

$\displaystyle\qquad\qquad\qquad\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})\underset{n\to\infty}{\to}\tilde{c}(x,y)\quad\text{where}$
(18)
$\displaystyle\tilde{c}(x,y)\,x^{1-\alpha}<y^{\beta},\;\;\tilde{c}(x,y)\underset{y\to
0}{\sim}y^{-\beta},\;\;\tilde{c}(x,y)=o\left(y^{-\beta}\right)\;\text{for}\;y\to\infty,$
$\displaystyle\text{with}\quad\tilde{a}_{n}=\mathcal{O}\left(n^{\frac{1-\alpha}{\beta}}\right)\quad\text{
and }\quad\tilde{b}_{n}={O}\left(\tilde{a}_{n}\right)\text{ or
}\,o\left(\tilde{a}_{n}\right).$

Then, the joint distribution of the concomitants maxima $(V_{1},V_{2})$
defined in (1) satisfies:

$\begin{split}&F_{(V_{1},V_{2})}(\tilde{a}_{n}v_{1}+\tilde{b}_{n},\,nv_{2})\\\
&\xrightarrow[n\to\infty]{}\int_{0}^{+\infty}H_{1}^{k}(v_{1}\mid
x)\,H_{2}(v_{2}\mid
x)\,\frac{x^{-k-2}}{k!}\,\emph{{e}}^{-1/x}\,\textrm{d}{x},\quad\forall
v_{1},v_{2}\in\mathbb{R},\end{split}$ (19)

where $H_{1}(y\mid x):=1-\tilde{c}(x,y)\,x^{1-\alpha}\,y^{-\beta}$ and
$\displaystyle H_{2}(y\mid x)$ is defined by the product of the limits given
below in (20) and (21), respectively, with
$r(x,y)=\lambda_{u}^{-1}\tilde{c}(x,y)$.

To prove Theorem 1, we need to evaluate the asymptotic behavior of the three
conditional distributions $F_{1}$, $F_{2}$ and $F_{3}$ defined in (3), (4) and
(5), respectively, each adequately transformed to obtain a non-degenerated
limit distribution. This is what is presented in the following lemma.

######  Lemma 2.

Assume the LT joint-tail model (15) holds and call $r$ the limit function
corresponding to $\mathcal{L}$. Then, as $n\to\infty$,

$\displaystyle F^{n}_{2}(ny\mid nx)$
$\displaystyle\to\exp\left\\{-\frac{1}{y}\left(1-\lambda_{u}r(x,y){\left(\frac{y}{x}\right)}^{\alpha}\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}\right)\right\\},$ (20) $\displaystyle
F_{3}(ny\mid nx)$ $\displaystyle\to
1+\lambda_{u}x^{-\alpha+2}y^{\alpha-1}\left(\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}r(x,y)-\frac{\alpha}{x}r(x,y)\right)\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}.$ (21)

Moreover, under Condition (18), we have

$\displaystyle F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid nx)$
$\displaystyle\underset{n\to\infty}{\to}\,\tilde{c}(x,y).$ (22)

Note that taking $\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}=0$ in (20) and (21) provides
$\displaystyle\lim_{n\to\infty}F^{n}_{2}(ny\mid nx)=\textrm{e}^{-1/y}$ and
$\displaystyle\lim_{n\to\infty}F_{3}(ny\mid nx)=1$, which characterise copulas
with asymptotic independence, as considered in Joshi and Nagaraja (1995). The
proofs of Theorem 1 and Lemma 2 are developed in Appendix A.

It is worth noticing that Condition (18) is needed to ensure that
$H_{1}(v_{1}\mid x)$ is a proper cdf for $v_{1}\in(0,+\infty)$. Moreover, our
Theorem 1 gives precise formulations on the choice of the normalizing
constants $\tilde{a}_{n}$ and $\tilde{b}_{n}$. This is in fact a delicate
point, reason why we dedicate a comprehensive discussion on it: Explanations
relative to Condition (18) are given in the next subsection, while Section 4
addresses the fundamental example of the Gaussian bivariate distribution, for
which Condition (18) does not hold.

###  2.3 Discussion on the choice of $\tilde{a}_{n}$ and $\tilde{b}_{n}$

In the following, we carefully deduce which are the only possible asymptotic
behaviours of the constants $\tilde{a}_{n}$ and $\tilde{b}_{n}$. We highlight
the importance of the present subsection, as finding suitable constants which
lead to non degenerate limits represents generally an issue in asymptotic
theories.

We start by noticing that the limit of $F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid
nx)$ strongly depends on that of
$\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})$. Indeed, we can write

$\displaystyle F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid nx)$
$\displaystyle=\,1-\frac{\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n}){(nx)}^{-\alpha}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{-\beta}}{1-\textrm{e}^{-1/(nx)}}$
$\displaystyle\underset{n\to\infty}{\sim}\,1-\frac{\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}$
(23)

using the approximation
$1-\textrm{e}^{-1/(nx)}\underset{n\to\infty}{\sim}{(nx)}^{-1}$.

In the following, we discuss which sequences $\tilde{a}_{n}$ and
$\tilde{b}_{n}$ should be chosen to have
$F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid nx)\underset{n\to\infty}{\to}1$,
_i.e._ $V_{1}$ not degenerate.

Given a couple $(x,y)$, we study the limit of (2.3) depending on the behavior
of $\mathcal{L}$, namely:

  1. (i)

Assume $\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})\to 0$ as $n\to\infty$. To
have $F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid nx)\not\to 1$, we need
$\frac{\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}\not\to
0$. But, since we are in the case
$\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})\underset{n\to\infty}{\to}0$,
this would mean, either

     * •

$n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}\underset{n\to\infty}{\to}0$
at the same rate of $\mathcal{L}$, which is impossible because of the
assumptions of the model ($\mathcal{L}\sim\text{SV}$), or

     * •

$n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}\underset{n\to\infty}{\to}0$
faster than $\mathcal{L}$, which is impossible because we would have
$\frac{\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}\underset{n\to\infty}{\to}\infty$,
that denies $F_{1}$ is a cdf.

Hence case (i) is not possible.

  2. (ii)

Assume $\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})\to\infty$ as
$n\to\infty$. So, as $n\to\infty$, the term
$n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}$ tends to
infinity faster than $\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})$ does,
which yields that $F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid
nx)\underset{n\to\infty}{\to}1$. Hence, this case (ii) has also to be
discarded.

  3. (iii)

The previous reasoning implies that we can only have, as $n\to\infty$,

$\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})\underset{n\to\infty}{\to}\tilde{c}(x,y)\in(0,\infty).$

Which conditions on $\tilde{a}_{n}$ and $\tilde{b}_{n}$ can be given? Here, we
have to distinguish whether $\tilde{c}(x,y)$ depends on $y$ or not.

    1. (A)

_Suppose $\tilde{c}(x,y)$ depends on $y$._ Then,
$\displaystyle\frac{\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}\underset{n\to\infty}{\sim}\frac{\tilde{c}(x,y)}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}$,
so we have to consider the relative behaviour of $\tilde{a}_{n}$ and
$\tilde{b}_{n}$.

      1. (a)

If $\tilde{a}_{n}=o(\tilde{b}_{n})$, then
$\tilde{a}_{n}y+\tilde{b}_{n}\,{\sim}\,\tilde{b}_{n}$, as $n\to\infty$. This
implies
$\frac{\tilde{c}(x,y)}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}\underset{n\to\infty}{\sim}\frac{\tilde{c}(x,y)}{n^{\alpha-1}x^{\alpha-1}{\tilde{b}_{n}}^{\beta}}$,
so, in order to maintain the dependence on $y$, we need
$n^{\alpha-1}\tilde{b}_{n}^{\beta}\underset{n\to\infty}{\not\to}0,\infty$.
This condition obviously depends on the value of $\alpha$: Call
$\gamma=(1-\alpha)/\beta$, then

         * •

If $0<\alpha<1$, $n^{\alpha-1}\underset{n\to\infty}{\to}0$, the only
possibility is $\tilde{b}_{n}=\mathcal{O}\left(n^{\gamma}\right)$ with
$\gamma\in(0,1)$, since $\alpha+\beta\geq 1$.

         * •

If $\alpha=1$, then $\tilde{b}_{n}$ would be independent of $n$; it is then
impossible to propose $(\tilde{a}_{n},\tilde{b}_{n})$ in such a case.

         * •

If $\alpha>1$, then $\tilde{b}_{n}=\mathcal{O}\left(n^{\gamma}\right)$ with
$\gamma<0$. However, since we are interested in the dependence in the
extremes, we need to consider

$n\mathbb{P}\\!\left(X>nx,Y>n^{\gamma}y\right)\leq n^{1-\gamma}\cdot
n^{\gamma}\mathbb{P}\\!\left(X>n^{\gamma}x,Y>n^{\gamma}y\right)\underset{n\to\infty}{\not\to}0,$

which implies $\gamma\in(0,1)$. So, the case $\alpha>1$ cannot be considered.

      2. (b)

If $\tilde{b}_{n}=o\left(\tilde{a}_{n}\right)$ or
$\tilde{b}_{n}=\mathcal{O}\left(\tilde{a}_{n}\right)$, then
$\tilde{a}_{n}y+\tilde{b}_{n}\underset{n\to\infty}{\sim}\tilde{a}_{n}y$ and

$\frac{\tilde{c}(x,y)}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}\underset{n\to\infty}{\sim}\frac{1}{n^{\alpha-1}{\tilde{a}_{n}}^{\beta}}\,\tilde{c}(x,y)\,x^{1-\alpha}.$

Hence, we need to consider $\tilde{a}_{n}=\mathcal{O}\left(n^{\gamma}\right)$
and $\alpha\in(0,1)$.

    2. (B)

_Suppose $\tilde{c}(x,y)$ does not depend on $y$._ In this case,

$\frac{\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})}{n^{\alpha-1}x^{\alpha-1}{(\tilde{a}_{n}y+\tilde{b}_{n})}^{\beta}}\underset{n\to\infty}{\sim}\frac{n^{1-\alpha}}{\max\left(\tilde{a}_{n}^{\beta}y^{\beta},\tilde{b}_{n}^{\beta}\right)}\tilde{c}x^{1-\alpha},$

and since we do not want to loose the dependence on $y$, we need to assume
$\max\left(\tilde{a}_{n}^{\beta}y^{\beta},\tilde{b}_{n}^{\beta}\right)=\tilde{a}_{n}^{\beta}y^{\beta}$.
Moreover, we need to ask $n^{1-\alpha}\tilde{a}_{n}^{-\beta}\not\to 0$
(otherwise $F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid
nx)\underset{n\to\infty}{\to}1$), which implies
$\tilde{a}_{n}=\mathcal{O}\left(n^{\gamma}\right)$ and $\alpha\in(0,1)$.

Therefore, to obtain a non degenerate limit for
$F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid nx)$, we need to assume
$\alpha\in(0,1)$, $\tilde{a}_{n}=\mathcal{O}\left(n^{\gamma}\right)$ and
$\tilde{b}_{n}=o\left(\tilde{a}_{n}\right)$ or
$\tilde{b}_{n}=\mathcal{O}\left(\tilde{a}_{n}\right)$. Now that the main
result is provided with a justification of the constants involved, let us
illustrate the theoretical results on standard examples of the risk
literature.

##  3 Illustration of the results

In this section, we consider two examples: the Pareto - Lomax with Survival
Clayton copula and the Bivariate Symmetric Logistic Extremal copula. We apply
our asymptotic approximations given in Theorem 1 and compare it with the
empirical results obtained via simulation. The choice of such examples lies in
the fact that the extremal dependence is ruled by parameters, so easy to
control. Note also that for the following examples we can achieve the same
results by direct computation (with a considerable amount of time and tedious
calculations), thus it is possible to make comparisons by means of the
mathematical expressions obtained.

We simulate $5000$ replicates of $100000$ bivariate samples. The dimension $k$
of the subsample used to compute $V_{1}$ is set to $10$ (Appendix B contains
simulations relative to other values of $k$, to have insights of its impact on
the asymptotic result).

###  3.1 Pareto - Lomax marginals with Survival Clayton copula

Let $(X_{P},Y_{P})$ be a random vector with Survival Clayton$(\theta)$ copula
and Pareto - Lomax$(\nu,1)$ marginals (the subscript $P$ indicates the Pareto
marginals):

$\displaystyle\bar{F}_{X_{P}}(x)={(1+x)}^{-\nu},\,\forall
x>0,\quad\bar{F}_{Y_{P}}(y)={(1+y)}^{-\nu},\,\forall y>0,\quad\nu>0,$
$\displaystyle\bar{F}_{P}(x,y)=\mathbb{P}\\!\left(X_{P}>x,Y_{P}>y\right)={\left({(1+x)}^{\nu\theta}+{(1+y)}^{\nu\theta}-1\right)}^{-1/\theta},\,\theta>0.$

(a)  (b)

Figure 1: Joint tail $\bar{F}(x,y)$ (gray) with unit Fréchet marginals, as
expressed in (24), and its approximation (black and gold) given by Equation
(15). The parameters $\alpha$ and $\beta$ are both set to $0.5$ in the two
examples.

First, we transform te Pareto marginals into the unit Fréchet ones, to match
the assumptions of the model given in Theorem 1. We can write

$F(x,y)=F_{P}\left(F_{X_{P}}^{-1}\left(\textrm{e}^{-1/x}\right),F_{Y_{P}}^{-1}\left(\textrm{e}^{-1/y}\right)\right),$

where, for the Pareto distribution,
$F_{X_{P}}^{-1}(q)=F_{Y_{P}}^{-1}(q)={(1-q)}^{-1/\nu}-1$. We obtain

$\bar{F}(x,y)={\left({\left(1-\textrm{e}^{-1/x}\right)}^{-\theta}+{\left(1-\textrm{e}^{-1/y}\right)}^{-\theta}-1\right)}^{-1/\theta}.$
(24)

Now, observe that the joint survival (24) can be rewritten, for $x>x_{0}>0$
and $y>y_{0}>0$, as

$\begin{split}\bar{F}(x,y)&\sim{\left(x^{\theta}+y^{\theta}-1\right)}^{-1/\theta}={\left(x^{\theta}+y^{\theta}-1\right)}^{-1/\theta}{(xy)}^{1/2}{(xy)}^{-1/2}\\\
&=\mathcal{L}(x,y)x^{-1/2}y^{-1/2},\end{split}$

where
$\mathcal{L}(x,y)={\left(x^{\theta}+y^{\theta}-1\right)}^{-1/\theta}{(xy)}^{1/2}{(xy)}$
and clearly $\alpha=\beta=1/2$.

Note that

$\mathcal{L}(n,n)={(2n^{\theta}-1)}^{-1/\theta}n\to 2^{-1/\theta}\quad\text{as
}n\to\infty.$

This is coherent with the value of the upper-tail dependence coefficient,
since

$\begin{split}\lambda_{U}=\lim_{q\to 1}\mathbb{P}\\!\left(Y>F_{Y}^{-1}(q)\mid
F_{X}^{-1}(q)\right)=\lim_{q\to
1}{(1-q)}^{-1}{(2{(1-q)}^{-\theta}-1)}^{-1/\theta}=2^{-1/\theta}.\end{split}$

(a)  (b)  (c)

Figure 2: Survival Clayton$(\theta=2)$, with $k=10$. Panel (a) shows the
empirical univariate log cdf of $V_{1}$ (in black), together with the
asymptotic theoretical log values (in gold) obtained by numerically
integrating the asymptotic joint cdf (19) to obtain the marginal. Panel (b)
shows the absolute values of the errors between empirical and theoretical cdf
of $V_{2}$. The upper plot of panel (c) shows the empirical univariate cdf of
$V_{1}$ (in black), together with the asymptotic theoretical values (in gold),
in log scale, obtained by numerically integrating the asymptotic joint cdf
(19). The lower plot of panel (c) represents the $L_{1}$ error between the
marginal empirical and theoretical cdf.

According to Theorem 1, since $1-\alpha=\beta$, we have
$\tilde{a}_{n}=a_{n}=n$. Hence,

$\mathcal{L}(nx,ny)={\left(n^{\theta}x^{\theta}+n^{\theta}y^{\theta}-1\right)}^{-1/\theta}n{(xy)}^{1/2}\underset{n\to\infty}{\to}{\left(x^{\theta}+y^{\theta}\right)}^{-1/\theta}{(xy)}^{1/2}=\tilde{c}(x,y).$

Note that, for each $a>0$,

$\tilde{c}(ax,ay)={\left({(ax)}^{\theta}+{(ay)}^{\theta}\right)}^{-1/\theta}{(a^{2}xy)}^{1/2}=\tilde{c}(x,y),$

which means that $\tilde{c}(x,y)$ and also
$r(x,y):=2^{1/\theta}\tilde{c}(x,y)$ satisfy (11). Moreover it is
straightforward to show that $\tilde{c}(x,y)$ satisfies Condition (18) in
Theorem 1. Since it holds

$\frac{\partial}{\partial x}r(x,y)=2^{1/\theta}\frac{\partial}{\partial
x}\tilde{c}(x,y)=2^{1/\theta}x^{-1}\tilde{c}(x,y)\left(\frac{1}{2}-{\left(1+{\left(\frac{y}{x}\right)}^{\theta}\right)}^{-1}\right),$

applying Lemma 2 and Theorem 1 gives that

$\displaystyle H_{1}(y\mid x)$
$\displaystyle=1-{\left(1+{\left(\frac{y}{x}\right)}^{\theta}\right)}^{-1/\theta},$
(25a) $\displaystyle H_{2}(y\mid x)$
$\displaystyle=\left(1-{\left(1+{\left(\frac{y}{x}\right)}^{\theta}\right)}^{-1/\theta-1}\right)\textrm{e}^{-\frac{1}{y}+{\left(1+{\left(\frac{y}{x}\right)}^{\theta}\right)}^{-1/\theta}}.$
(25b)

The plot in Figure LABEL:subfig:_SC_comparison_surface shows the bivariate
surface computed by applying Theorem 1 (the integration is performed
numerically) and the values of the bivariate empirical cdf computed in the
simulated $V_{1}$ and $V_{2}$. These points lie on the surface, thus
displaying the very good quality of the asymptotic approximation of Theorem 1.
It can also be assessed from Figures LABEL:subfig:_SC_comparison_boxplot and
LABEL:subfig:_SC_comparison_margins, showing the boxplot of the absolute value
of the errors between the black cloud and the surface in Figure
LABEL:subfig:_SC_comparison_surface, the asymptotic approximation of the
marginal distributions of $V_{1}$ and the errors in $V_{2}$, respectively. For
what concerns the error (lower plot of Figure
LABEL:subfig:_SC_comparison_margins), we observe that it is small for low
values of $v_{2}$ (the maximum error is close to $1\%$) and converges very
quickly to $0$. In particular, it is $0\%$ for all the values of $v_{2}$ we
are considering.

###  3.2 Unit Fréchet marginals with Logistic Extremal copula

Let $(X,Y)$ be a random vector with unit Fréchet marginals and a bivariate
symmetric logistic extremal dependence, _i.e._

$F(x,y)=\textrm{e}^{-V(x,y)},$

where

$V(x,y)={\left(x^{-1/\gamma}+y^{-1/\gamma}\right)}^{\gamma},\,x,y>0,\,\gamma\in(0,1).$

By considering the bivariate joint tail approximation (15) proposed in Ledford
and Tawn (1997) characterised by a general extremal dependence function $V$,
we can write in our case

$\mathcal{L}(x,y)={(xy)}^{-1/2}{(x+y)}-{(xy)}^{1/2}V(x,y).$ (26)

(a)  (b)  (c)

Figure 3: Logistic$(\gamma=1/2)$, with $k=10$. Panel (a) shows the empirical
univariate log cdf of $V_{1}$ (in black), together with the asymptotic
theoretical log values (in gold) obtained by numerically integrating the
asymptotic joint cdf (19) to obtain the marginal one. Panel (b) shows the
absolute values of the errors between the empirical and the theoretical cdf of
$V_{2}$. The upper plot of panel (c) shows the empirical univariate cdf of
$V_{1}$ (in black), together with the asymptotic theoretical values (in gold),
in log scale, obtained by numerically integrating the asymptotic joint cdf
(19). The lower plot of panel (c) represents the $L_{1}$ error between the
marginal empirical and theoretical cdf.

Note that

$\mathcal{L}(n,n)=2-V(1,1),$

since $V$ is homogeneous of order $-1$ (_i.e._ $V(nx,ny)=n^{-1}V(x,y)$).
Moreover, $1-\alpha=\beta$, so that $\tilde{a}_{n}=a_{n}=n$. Hence, we can
write

$\begin{split}\mathcal{L}(nx,ny)&=n^{-1}{(xy)}^{-1/2}n(x+y)-n{(xy)}^{1/2}n^{-1}V(x,y)\\\
&={(xy)}^{-1/2}{(x+y)}-{(xy)}^{1/2}V(x,y)=\tilde{c}(x,y).\end{split}$

Note that, for each $a>0$,

$\tilde{c}(ax,ay)=\tilde{c}(x,y),$

so that the candidate $\mathcal{L}$ in (26) is a proper BSV function. Simple
algebra helps to prove that $\tilde{c}(x,y)$ satisfies Condition (18) in
Theorem 1. Since it holds

$\frac{\partial}{\partial
x}\tilde{c}(x,y)=\frac{{(xy)}^{-1/2}}{2}\left(1-\frac{y}{x}-yV(x,y)\left(1-2{(xV(x,y))}^{-1/\gamma}\right)\right),$

by applying Lemma 2 and Theorem 1, we obtain

$\displaystyle H_{1}(y\mid x)$ $\displaystyle=x\left(V(x,y)-y^{-1}\right),$
(27a) $\displaystyle H_{2}(y\mid x)$
$\displaystyle={\left(1+{\left(\frac{y}{x}\right)}^{-\frac{1}{\gamma}}\right)}^{\gamma-1}\textrm{e}^{-\frac{1}{x}\left({\left(1+{\left(\frac{y}{x}\right)}^{-\frac{1}{\gamma}}\right)}^{\gamma}-1\right)}.$
(27b)

After having obtained the functions $H_{1}$ and $H_{2}$, we can now apply a
numerical integration to compute the joint cdf (19).

##  4 Discussion on the Gaussian case

Whilst our focus so far was on obtaining the joint asymptotic cdf of
$(V_{1},V_{2})$ when $(X,Y)$ exhibits asymptotic or extremal dependence, we
question if Condition (18) of Theorem 1 is strongly related to the extremal
dependence assumption. In other words, what does occur if the BSV function
$\mathcal{L}$ goes to $0$ or to infinity? This is why we return to the case
when $(X,Y)$ is Gaussian, as it is well known that $(X,Y)$ is then
asymptotically independent. This case has already been treated in the
literature, due both to practical interest of the distribution and to its
known manageability. Although direct computations in this case become simple,
problems arise in applying models involving approximations with SV functions.
In fact, even if such approximations are provided, they are then not used to
compute for instance the conditional distributions we encountered above (as
far as we could observe in the literature). The objective of the present
section is to question this apparent paradox and provide tools to overcome
this issue, thus bridging the gap between direct computations and those made
through asymptotic models, with some attention put on the asymptotic behaviour
of the BSV $\mathcal{L}$.

###  4.1 Existing literature

Let us start to review the existing literature on the Gaussian case.

  1. (a)

Joshi and Nagaraja (1995): By direct computations, the authors are able to
provide constants $\tilde{a}_{n},\,\tilde{b}_{n}$ such that the rescaled
distribution of $(V_{1},V_{2})$ converges to a product of a standard normal
cdf to the power $k$ and a Gumbel cdf. When applying those constants in our
case, after the transformation of the Gaussian marginals into unit Fréchet,
Condition (18) in Theorem 1 is not satisfied. In fact, we obtain that $F_{1}$
degenerates for $n\to\infty$.

Hence, our question: Could we find other transformations of the marginals that
let us circumvent the case $\mathcal{L}\to 0$, and possibly find a new method
or solution to handle this case? The relevance of this problem lies in the
fact that relations between asymptotic dependence and the behavior of
$\mathcal{L}$ are generally unknown: We know by formula (16) that the
behaviour of $\mathcal{L}(n,n)$ and the values $\eta$ takes are linked to the
upper-tail dependence coefficient $\lambda_{u}$, but the case
$\mathcal{L}\not\to 0$ and $\eta<1$ is not tackled. Moreover, we also need to
consider $\mathcal{L}(n,\tilde{a}_{n})$ instead of $\mathcal{L}(n,n)$.

  2. (b)

Ledford and Tawn (1998): The focus here is only on the asymptotic distribution
of the concomitant of the maximum $X_{(n)}$, that is the asymptotic
distribution of $V_{1}$ when $k=1$ (in our notation), meaning to study
$F_{1}$. The authors explain that there can be cases where unit Fréchet
marginals are not appropriate and suggest a standard Gumbel transformation for
the variable $Y$, arriving to a new BSV function $\mathcal{L}_{1}$ (in fact
the first order approximation of $\mathcal{L}$) that avoids the degeneracy of
the limit of $H_{1}$ (with our notations). However, to compare this approach
with the case where both the marginals are unit Fréchet, they need to assume
$y\to\infty$.

Let $(X_{N},Y_{N})$ be a bivariate normal random vector with standard normal
marginals and with correlation $0<\rho<1$. Theorem 3 in Joshi and Nagaraja
(1995) states that, in the extreme case ($k$ fixed, $n\to\infty$), it holds
that

$\mathbb{P}\\!\left(\frac{V_{1}-\tilde{a}_{n}}{\tilde{b}_{n}}\leq
v_{1},\frac{V_{2}-a_{n}}{b_{n}}\leq
v_{2}\right)\underset{n\to\infty}{\to}\Phi^{k}(v_{1})\textrm{e}^{-\textrm{e}^{-v_{2}}},$
(28)

where

$\displaystyle a_{n}$ $\displaystyle={\left(2\log n\right)}^{-1/2}$
$\displaystyle b_{n}$ $\displaystyle=\sqrt{2\log
n}-\frac{1}{2}\frac{\log\left(4\pi\log n\right)}{\sqrt{2\log n}}$ (29a)
$\displaystyle\tilde{a}_{n}$ $\displaystyle=\sqrt{1-\rho^{2}}$
$\displaystyle\tilde{b}_{n}$ $\displaystyle=\rho b_{n}.$ (29b)

Formula (28) represents an asymptotic independence property of $V_{1}$ and
$V_{2}$, strongly connected to the fact that

$F_{1}(\tilde{a}_{n}y+\tilde{b}_{n}\mid
a_{n}x+b_{n})=\mathbb{P}\\!\left(Y_{N}\leq\tilde{a}_{n}y+\tilde{b}_{n}\mid
X_{N}>a_{n}x+b_{n}\right)\underset{n\to\infty}{\to}\Phi(y).$ (30)

Recall that our model in Theorem 1 is not applicable here. In fact, Condition
(18) is not satisfied as $\mathcal{L}(n,\tilde{a}_{n})\to 0$ for any choice of
the normalizing constants. Thus, further study would be needed in such a case.

A way to look at the problem may be through Wadsworth and Tawn (2013). The
authors study the effect of letting the components of a bivariate random
vector grow at different rates, with the result of stating a new class of
regular variation conditions and providing a new characterization of the link
between the multivariate tail decay and the considered marginal growth rates.

More precisely, the authors transform the marginals of the original vector
into standard exponentials, called $(X_{E},Y_{E})$, and assume that, for all
$(\zeta_{1},\zeta_{2})\in\mathbb{R}_{+}^{2}\setminus\\{\bm{0}\\}$,

$\mathbb{P}\\!\left(X_{E}>\zeta_{1}\log n,Y_{E}>\zeta_{2}\log
n\right)\underset{n\to\infty}{\sim}L(n;\zeta_{1},\zeta_{2})n^{-\kappa(\zeta_{1},\zeta_{2})}$
(31)

where $L$ is a univariate SV function in $n$ as $n\to\infty$, and the function
$\kappa(\zeta_{1},\zeta_{2})$ _maps the different marginal growth rates to the
joint tail decay rate_ with the role of providing information _about the level
of dependence between variables at sub-asymptotic levels_. Assumption (31) is
then used to describe models for conditional probabilities of extreme
exceedances, _i.e._

$\mathbb{P}\\!\left(X_{E}>\zeta_{1}\log n\mid Y_{E}>\log
n\right)\underset{n\to\infty}{\sim}L(n;\zeta_{1},1)\textrm{e}^{(1-\kappa(\zeta_{1},\zeta_{2}))\log
n}.$ (32)

Studying (32) is proven by the authors to be equivalent to the search for
functions $a^{*},\,b^{*}$ and $h$ such that

$\lim_{n\to\infty}\mathbb{P}\\!\left(\frac{X_{E}-b^{*}(\log n)}{a^{*}(\log
n)}>x\mathrel{}\middle|\mathrel{}Y_{E}>\log n\right)=\textrm{e}^{-h(x)}.$ (33)

The use of the exponential marginals is helpful to identify the normalizing
constants for the bivariate Gaussian example, thus partially filling the gap
left open in Ledford and Tawn (1998), where, as previously explained, a way to
tackle the problem is given in the case of Fréchet-Gumbel marginals, albeit
that solution can be used only for some high values of $Y$. However, this
discussion does not yet answer our main question: Why, when using unit Fréchet
marginals, aren’t we able to find normalizing constants that lead to a non-
degenerate limit for $\mathcal{L}(nx,\tilde{a}_{n}y+\tilde{b}_{n})$? Our
understanding of the problem is made more complete by the insightful paper by
Heffernan and Resnick (2007). There, the authors explore the implications of
assuming the existence of a scaling function $c^{*}_{1}(\cdot)$, a centering
function $c^{*}_{2}(\cdot)$, and a non-null Radon measure $\mu$ on Borel
subsets of $[-\infty,\infty]\times(-\infty,\infty]$ such that, for each fixed
$y\in E_{\xi}:=\\{y\in\mathbb{R}:1+\xi y>0\\}$ (where $\xi$ is the shape
parameter of the extreme value distribution relative to $F_{Y}$), it holds

  1. (a)

$\mu([-\infty,x]\times(y,+\infty))$ is a finite and non-degenerate
distribution function in $x$,

  2. (b)

the limit of the cdf of the normalized $X$ given that $Y$ is extreme exists
and

$\mathbb{P}\\!\left(\frac{X-c^{*}_{2}\circ b^{\leftarrow}(t)}{c^{*}_{1}\circ
b^{\leftarrow}(t)}\leq x\mathrel{}\middle|\mathrel{}Y>t\right)\underset{t\to
y^{+}}{\to}\mu([-\infty,x]\times(0,+\infty)),$ (34)

where $y^{+}$ is the right-end point of $F_{Y}$. It is worth noticing the fact
that we are free to change the marginal of $Y$ without disturbing the
convergence in (34), but this is not true for the $X$ variable. In the paper
it is proven that, in the case of the bivariate standard Gaussian, it is not
possible to obtain a non-degenerate expression for $\mu$ in (34) if the
marginals of $X$ and $Y$ are both transformed into the Pareto type, while an
asymptotic form is still reachable if the marginals are transformed into
exponentials. We refer to (Heffernan and Resnick, 2007, Sec. 7) for a thorough
discussion of this aspect.

###  4.2 Revisiting the Gaussian case

Now that we have a better and more complete understanding on the issues
underlying the treatment made in the literature of the Gaussian case, we
provide a way to find suitable constants through the application of model (32)
and formulas (29a) and (29b). We know that, if $(X_{N},Y_{N})$ is a bivariate
normal random vector and correlation $\rho\in(0,1)$,

$\mathbb{P}\\!\left(Y_{N}>\tilde{a}_{n}y+\tilde{b}_{n}\mid
X_{N}>a_{n}x+b_{n}\right)\underset{n\to\infty}{\to}1-\Phi(y),$ (35)

where the constants are given in (29a) and (29b). The following results hold:

######  Theorem 2.

Let $(X_{E},Y_{E})$ be a bivariate random vector with unit exponential
marginals and correlation $\rho\in(0,1)$, such that (32) holds, with
$\zeta_{1}:=\zeta_{1}(n,y)$. Then, we have

$\mathbb{P}\\!\left(Y_{E}>\tilde{a}_{n,E}y+\tilde{b}_{n,E}\mid X_{E}>\log
n\right)\to\frac{1}{y\sqrt{2\pi}}\textrm{e}^{-{(\rho y)}^{2}/2},\quad\text{as
}n\to\infty,$ (36)

where

$\displaystyle\tilde{a}_{n,E}$
$\displaystyle=\rho\sqrt{1-\rho^{2}}\left(\sqrt{2\log
n}-\frac{1}{2}\frac{\log\left(4\pi\log n\right)}{\sqrt{2\log n}}\right)$ (37a)
$\displaystyle\text{and\hskip 14.22636pt}\tilde{b}_{n,E}$
$\displaystyle=\rho^{2}\log n-\frac{\rho^{2}}{2}\log\left(4\pi\log
n\right)+\frac{\rho^{2}}{16}\frac{{(\log\left(4\pi\log n\right))}^{2}}{\log
n}.$ (37b)

######  Corollary 2.

Note that, if we assume $y$ large enough (as in Ledford and Tawn (1998)), then
we can use the Mill’s ratio to obtain

$\mathbb{P}\\!\left(Y_{E}>\tilde{a}_{n,E}y+\tilde{b}_{n,E}\mid X_{E}>\log
n\right)\underset{n\to\infty}{\sim}\rho\left(1-\Phi(\rho y)\right).$ (38)

Refer to Appendix A for the proofs.

The limit distribution given in (38) is similar to those in Joshi and Nagaraja
(1995) and Wadsworth and Tawn (2013). The differences among the final formulas
are due to the different chosen orders of the Taylor approximations.

##  5 Discussion

The work presented has provided a deep understanding of the impact of the
extremal dependence of a bivariate random vector $(X,Y)$ on the joint
asymptotic behavior of suitably rescaled maxima over subsets of concomitants.
Studying the framework of asymptotically dependent random variables is
actually of great interest in practical applications and completes the
literature.

Our main result addresses this problem, providing the expression of the
asymptotic joint cdf of the two rescaled maxima of concomitants. In
particular, we take advantage of the joint-tail model proposed in Ledford and
Tawn (1997) to provide the asymptotic form of all the conditional
distributions describing $Y$ given the behaviour of $X$, which contribute to
the asymptotic joint cdf of the two maxima of concomitants. We also explain in
detail how to choose suitable normalizing constants needed to avoid the
degeneracy of such distributions (thus preventing the joint cdf of the maxima
of concomitants to be degenerate as well). Our theoretical results are
illustrated through two examples of interest in the risk analysis literature:
We compute the analytical expressions of the conditional distribution and
perform a numerical integration to obtain the joint cdf. We show the goodness
of the asymptotic approximation in all the examined cases through a numerical
simulation.

Furthermore, we reconsider Theorem 2 in Joshi and Nagaraja (1995) and show how
some hypotheses are connected to the upper-tail dependence coefficient
characterizing the copula of the parent vector $(X,Y)$, thus providing an
intuitive interpretation of the formulas therein and a new reformulation of
the cited Theorem.

More importantly, we revisit the example of the bivariate Gaussian
distribution in our case, furnishing a comprehensive discussion on it. We
deeply study the reasons why the joint-tail model in Ledford and Tawn (1997)
fails in providing a non-degenerate asymptotic distribution and fill the gap
in the extension of the joint-tail model proposed in Wadsworth and Tawn
(2013), by showing how to compute suitable normalization constants in this
case. Our results are coherent with those obtained by direct computation in
Joshi and Nagaraja (1995).

##  Appendix A Appendix - Proofs

###### Proof of Lemma 1.

Using the notations of Lemma 1, and indicating with
$\mathscr{D}(G_{\gamma_{1}})$, $\mathscr{D}(G_{\gamma_{2}})$,
$\mathscr{D}(G_{0})$ the MDAs of the Fréchet, the Weibull and the Gumbel
distributions, respectively, we can write, for $x,y\in\mathbb{R}$ and $x^{+}$
denoting the upper end-point of the distribution of $X$,

$\begin{split}&n\mathbb{P}\\!\left(X>a_{n}x+b_{n},Y>c_{n}y+d_{n}\right)\\\
&=\int_{a_{n}x+b_{n}}^{x^{+}}n\mathbb{P}\\!\left(Y>c_{n}y+d_{n}\mid
X=t\right)f_{X}(t)\,\textrm{d}{t}\\\ &\leq
n\int_{a_{n}x+b_{n}}^{x^{+}}f_{X}(t)\,\textrm{d}{t}=n(1-F_{X}(a_{n}x+b_{n}))<\infty,\,\forall
n\in\mathbb{N}.\end{split}$

As shown in the proof of Proposition 2.5 in Resnick (1987), it holds

$na_{n}f_{X}(a_{n}z+b_{n})\mathds{1}\left(x\leq
z<a_{n}^{-1}(x^{+}-b_{n})\right)\underset{n\to\infty}{\to}\mathds{1}\left(z\geq
x\right)g(z)$

with

$g(z):=\begin{cases}\gamma_{1}z^{-(\gamma_{1}+1)},\,z>0,\,\gamma_{1}>0,&\text{if
}F_{X}\in\mathscr{D}(G_{\gamma_{1}}),\\\
\gamma_{2}(-z)^{\gamma_{2}-1},\,z<0,\,\gamma_{2}>0,&\text{if
}F_{X}\in\mathscr{D}(G_{\gamma_{2}}),\\\ e^{-z},\,z\in\mathbb{R},&\text{if
}F_{X}\in\mathscr{D}(G_{0}).\end{cases}$

Note the role of the indicator $\mathds{1}\left(x\leq
z<a_{n}^{-1}(x^{+}-b_{n})\right)$, needed to ensure that we are integrating
over the support of $X$. In order to apply the Extended Dominated Convergence
Theorem, we need to check that

$\int_{\mathbb{R}}na_{n}f_{X}(a_{n}z+x^{+})\mathds{1}\left(z\geq
x\right)\,\textrm{d}{z}=n\left(1-F_{X}(a_{n}x+b_{n})\right)\underset{n\to\infty}{\to}\int_{\mathbb{R}}g(z)\mathds{1}\left(z\geq
x\right)\,\textrm{d}{z}.$

For the Gumbel case, we have

$n\left(1-F_{X}(a_{n}x+b_{n})\right)\to\textrm{e}^{-x}\text{ and
}\int_{\mathbb{R}}g(z)\mathds{1}\left(z\geq
x\right)\,\textrm{d}{z}=\textrm{e}^{-x}.$

The Fréchet and Weibull cases follow the same reasoning. Since the two
quantities are equal, we can exchange limit and integral, as follows:

$\begin{split}&\lim_{n\to\infty}\int_{\mathbb{R}}n\mathbb{P}\\!\left(Y>c_{n}y+d_{n}\mid
X=t\right)f_{X}(t)\mathds{1}\left(t\geq a_{n}x+b_{n}\right)\,\textrm{d}{t}\\\
&=\int_{\mathbb{R}}\lim_{n\to\infty}\mathbb{P}\\!\left(Y>c_{n}y+d_{n}\mid
X=a_{n}z+b_{n}\right)\cdot\lim_{n\to\infty}na_{n}f_{X}(a_{n}z+b_{n})\\\
&\qquad\qquad\cdot\mathds{1}\left(z<a_{n}^{-1}(x^{+}-b_{n})\right)\cdot\mathds{1}\left(z\geq
x\right)\,\textrm{d}{z}.\end{split}$

Since the integrand is positive, then the integral tends to $0$ if and only if
the latter tends to $0$, that is, if and only if

$\mathbb{P}\\!\left(Y>c_{n}y+d_{n}\mid
X=a_{n}z+b_{n}\right)=1-F_{3}\left(c_{n}y+d_{n}\mid
a_{n}z+b_{n}\right)\underset{n\to\infty}{\to}0$

for all $z\geq x$ (note that
$\displaystyle{\lim_{n\to\infty}na_{n}f_{X}(a_{n}z+b_{n})\neq 0}$). Hence the
result. ∎

###### Proof of Lemma 2.

In order to make use of (15), we write all the conditionals in terms of
$\bar{F}$ and recall the notation $C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}$ for the case $\alpha+\beta=1\text{ and
}\lambda_{U}>0$. For what concerns $F_{2}(y\mid x)$, we have

$F_{2}(y\mid x)=\mathbb{P}\\!\left(Y\leq y\mid
X<x\right)=1-\mathbb{P}\\!\left(Y>y\mid
X<x\right)=1-\frac{\bar{F}_{Y}(y)-\bar{F}(x,y)}{F_{X}(x)}.$

Since

$\begin{split}&\frac{\bar{F}_{Y}(ny)-\bar{F}(nx,ny)}{F_{X}(nx)}=\textrm{e}^{\frac{1}{nx}}\left(1-\textrm{e}^{-\frac{1}{ny}}-\frac{\mathcal{L}(nx,ny)}{n^{\alpha+\beta}x^{\alpha}y^{\beta}}\right)\\\
&\underset{n\to\infty}{\sim}\frac{1}{n}\left(1+\frac{1}{nx}\right)\left(\frac{1}{y}-\frac{\mathcal{L}(nx,ny)}{n^{\alpha+\beta-1}x^{\alpha}y^{\beta}}\right)=\frac{1}{ny}\left(1+\frac{1}{nx}\right)\left(1-\frac{\mathcal{L}(nx,ny)}{n^{\alpha+\beta-1}x^{\alpha}y^{\beta-1}}\right)\\\
&\underset{n\to\infty}{\sim}\frac{1}{y}\left(1+\frac{1}{nx}\right)\left(1-\frac{\mathcal{L}(n,n)}{n^{\alpha+\beta-1}}r(x,ny)x^{-\alpha}y^{1-\beta}\right)\underset{n\to\infty}{\sim}\frac{1}{ny}\left(1-\lambda_{U}r(x,y){\left(\frac{y}{x}\right)}^{\alpha}\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}\right),\end{split}$

we obtain

$F^{n}_{2}(ny\mid
nx)={\left[1-\left(\frac{\bar{F}_{Y}(ny)}{F_{X}(nx)}-\frac{\bar{F}(nx,ny)}{F_{X}(nx)}\right)\right]}^{n}\underset{n\to\infty}{\sim}{\left[1-\frac{1}{ny}\left(1-\lambda_{U}r(x,y){\left(\frac{y}{x}\right)}^{\alpha}\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}\right)\right]}^{n}.$

Thus

$F^{n}_{2}(ny\mid
nx)\underset{n\to\infty}{\to}\textrm{e}^{-\frac{1}{y}\left(1-\lambda_{U}r(x,y){\left(\frac{y}{x}\right)}^{\alpha}\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}\right)}.$

For $F_{3}(y\mid x)$, note that

$\begin{split}F_{3}(y\mid
x)&=\frac{1}{f_{X}(x)}\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}F(x,y)=\frac{1}{f_{X}(x)}\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}\left(F_{X}(x)-\bar{F}(y)+\bar{F}(x,y)\right)\\\
&=1+\frac{1}{f_{X}(x)}\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}\bar{F}(x,y)=1+x^{2-\alpha}y^{-\beta}\textrm{e}^{\frac{1}{x}}\left(\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}\mathcal{L}(x,y)-\frac{\alpha}{x}\mathcal{L}(x,y)\right).\end{split}$

It follows that

$\begin{split}F_{3}(ny\mid
nx)&\underset{n\to\infty}{\sim}1+n^{2-\alpha-\beta}\textrm{e}^{\frac{1}{nx}}x^{2-\alpha}y^{-\beta}\left(\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial(nx)}\mathcal{L}(nx,ny)-\frac{\alpha}{nx}\mathcal{L}(nx,ny)\right)\\\
&=1+\textrm{e}^{\frac{1}{nx}}x^{2-\alpha}y^{-\beta}n^{2-\alpha-\beta}\left(\frac{1}{n}\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}\mathcal{L}(nx,ny)-\frac{\alpha}{nx}\mathcal{L}(nx,ny)\right)\\\
&\underset{n\to\infty}{\sim}1+\textrm{e}^{\frac{1}{nx}}n^{1-\alpha-\beta}\mathcal{L}(n,n)x^{2-\alpha}y^{-\beta}\left(\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}r(x,y)-\frac{\alpha}{x}r(x,y)\right)\\\
&\underset{n\to\infty}{\to}1+\lambda_{U}x^{2-\alpha}y^{-\beta}\left(\frac{\mathop{}\\!\partial}{\mathop{}\\!\partial
x}r(x,y)-\frac{\alpha}{x}r(x,y)\right)\mathds{1}_{C_{\not\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}},\end{split}$

which concludes the proof. ∎

###### Proof of Theorem 1.

The proof is based on Lemma 2, and Joshi and Nagaraja (1995). To begin with,
we report the finite-sample bivariate cumulative distribution of
$(V_{1},V_{2})$ ((Joshi and Nagaraja, 1995, Equation (2.1))),

$F_{(V_{1},V_{2})}(v_{1},v_{2})=\mathbb{E}\left[h\left(v_{1},v_{2},X_{(n-k)}\right)\right]$
(39)

where the function $h$ is defined as

$h(v_{1},v_{2},x)=F_{1}^{k}(v_{1}\mid x)F^{n-k-1}_{2}\left(v_{2}\mid
x\right)F_{3}\left(v_{2}\mid x\right).$

The rational under Equation (39) lies in Kaufmann and Reiss (1992).
Conditioned on the event $\\{X_{(n-k)}=x\\}$, the maxima $V_{1}$ and $V_{2}$
have the following behaviour: $V_{1}$ can be considered as the sample maximum
of a random sample of size $k$ from $F_{1}(\cdot\mid x)$, while $V_{2}$
behaves like the sample maximum of a sample of size $n-k$, composed by $n-k-1$
i.i.d. variables from $F_{2}(\cdot\mid x)$ and a single random variable from
$F_{3}(\cdot\mid x)$.

Following Joshi and Nagaraja (1995), from Equation (39), we can write

$\begin{split}F_{(V_{1},V_{2})}(\tilde{a}_{n}v_{1}+\tilde{b}_{n},\,c_{n}v_{2}+d_{n})=\int&F_{1}(\tilde{a}_{n}v_{1}+\tilde{b}_{n}\mid
a_{n}x+b_{n})F^{n-k-1}_{2}\left(c_{n}v_{2}+d_{n}\mid a_{n}x+b_{n}\right)\\\
&F_{3}\left(c_{n}v_{2}+d_{n}\mid
a_{n}x+b_{n}\right)a_{n}f_{X}(a_{n}x+b_{n})\,\textrm{d}{x}.\end{split}$

Since our goal is to compute the joint asymptotic distribution of $V_{1}$ and
$V_{2}$ as $n\to\infty$, we refer to Joshi and Nagaraja (1995) and Nagaraja
and David (1994) for what concerns the swap between the limit operation and
the integral. Since $X$ is unit Fréchet distributed, it is clear that the
marginal $F_{X}$ is in the domain of attraction of
$G_{X}(x)=\textrm{e}^{-1/x}$ and that $g_{X}(x)=x^{-2}\textrm{e}^{-1/x}$. We
refer to Lemma 1 and Result 1 in Nagaraja and David (1994), for the
convergence as $n\to\infty$ of the term $a_{n}f_{X}(a_{n}x+b_{n})$ to
$g_{(k)}(x):=\frac{{(-\log G_{X}(x))}^{k}}{k!}g_{X}(x)$. Theorem 1 then
follows by Condition (18) and Lemma 2. ∎

###### Proof of Theorem 2.

Note that $X_{N}>a_{n}x+b_{n}$ is equivalent to $X_{N}>b_{n}$, as $a_{n}\to 0$
when $n\to\infty$. Let us now report the model in Wadsworth and Tawn (2013) in
terms of standard normal marginals, namely

$\begin{split}&\mathbb{P}\\!\left(Y_{E}>\zeta_{1}(n,y)\log n\mid X_{E}>\log
n\right)\\\
&=\mathbb{P}\\!\left(Y_{N}>\Phi^{-1}\left(1-n^{-\zeta_{1}(n,y)}\right)\mathrel{}\middle|\mathrel{}X_{N}>\Phi^{-1}\left(1-n^{-1}\right)\right)\\\
&\sim
L_{1}\left(n;\zeta_{1}(n,y),1\right)n^{1-\kappa(\zeta_{1}(n,y),1)}\end{split}$
(40)

where

$\displaystyle\kappa(\zeta_{1}(n,y),1)=\frac{\zeta_{1}(n,y)+1-2\rho\sqrt{\zeta_{1}(n,y)}}{1-\rho^{2}},\qquad\text{and}$
(41) $\displaystyle L_{1}(n;\zeta_{1}(n,y),1)={(4\pi\log
n)}^{\frac{2\rho^{2}-\rho\left(\sqrt{\zeta_{1}(n,y)}+1/\sqrt{\zeta_{1}(n,y)}\right)}{2(1-\rho^{2})}}\frac{{\zeta_{1}(n,y)}^{\frac{1-\rho/\sqrt{\zeta_{1}(n,y)}}{2(1-\rho^{2})}}{(1-\rho^{2})}^{3/2}}{(\sqrt{\zeta_{1}(n,y)}-\rho)(1-\rho\sqrt{\zeta_{1}(n,y)})},$
(42)

whenever $\rho^{2}<\min\\{\zeta_{1}(n,y),1\\}$. The authors explain that the
SV function $L$, which we have in the case of the standard exponential
marginals (from Equation (32)), is such that $L/L_{1}\sim 1$ as $n\to\infty$.

To ease the notation, in the following, we use $\zeta_{1}$ for
$\zeta_{1}(n,y)$. We want to find some constants $\tilde{a}_{n,E}$ and
$\tilde{b}_{n,E}$ such that $\\{Y_{E}>\zeta_{1}(n,y)\log n\\}$ is equivalent
to $\\{Y_{E}>\tilde{a}_{n,E}y+\tilde{b}_{n,E}\\}$, so we set

$\zeta_{1}=\frac{\tilde{a}_{n,E}y+\tilde{b}_{n,E}}{\log n}.$ (43)

By using the first equality in (40) and knowing that if
$Y_{N}\sim\mathcal{N}(0,1)$ then
$-\log\left(1-\Phi(Y_{N})\right)\sim\text{Exp}(1)$, it follows that

$\zeta_{1}=\frac{-\log\left(1-\Phi(\tilde{a}_{n}y+\tilde{b}_{n}\right)}{\log
n}.$ (44)

This means that we can rewrite (35) by transforming the marginals into
standard exponential and then apply the model (40). Combining (43) and (44),
we obtain

$\begin{split}\tilde{a}_{n,E}y+\tilde{b}_{n,E}&=-\log\left(1-\Phi(\tilde{a}_{n}y+\tilde{b}_{n}\right)\\\
&\underset{n\to\infty}{\sim}-\log\left(\frac{\phi(\tilde{a}_{n}y+\tilde{b}_{n})}{\tilde{a}_{n}y+\tilde{b}_{n}}\right)\text{
through the Mill's ratio, if $\tilde{a}_{n}\to\infty$ or
$\tilde{b}_{n}\to\infty$}\\\
&=-\log\left(\frac{1}{\sqrt{2\pi}}\textrm{e}^{-{(\tilde{a}_{n}y+\tilde{b}_{n})}^{2}/2}\right)+\log\left(\tilde{a}_{n}y+\tilde{b}_{n}\right)\\\
&=\frac{1}{2}\log(2\pi)+\frac{1}{2}\tilde{a}_{n}^{2}y^{2}+\tilde{a}_{n}\tilde{b}_{n}y+\frac{1}{2}\tilde{b}_{n}^{2}+\log\tilde{b}_{n}+\log\left(1+\frac{\tilde{a}_{n}}{\tilde{b}_{n}}y\right)\\\
&\underset{n\to\infty}{\sim}\frac{1}{2}\log(2\pi)+\frac{1}{2}\tilde{a}_{n}^{2}y^{2}+\tilde{a}_{n}\tilde{b}_{n}y+\frac{1}{2}\tilde{b}_{n}^{2}+\log\tilde{b}_{n}+\frac{\tilde{a}_{n}}{\tilde{b}_{n}}y\quad\text{if
$\tilde{a}_{n}=o\left(\tilde{b}_{n}\right)$}\\\
&=\tilde{a}_{n}\left(\tilde{b}_{n}+\frac{1}{\tilde{b}_{n}}\right)y+\frac{1}{2}\log(2\pi)+\frac{1}{2}\tilde{a}_{n}^{2}y^{2}+\frac{1}{2}\tilde{b}_{n}^{2}+\log\tilde{b}_{n}\\\
&\underset{n\to\infty}{\sim}\tilde{a}_{n}\tilde{b}_{n}y+\frac{1}{2}\tilde{b}_{n}^{2}\quad\text{(the
discarded terms are $o\left(\tilde{a}_{n}\tilde{b}_{n}\right)$)}.\end{split}$

So, we can identify the terms, which gives

$\tilde{a}_{n,E}=\tilde{a}_{n}\tilde{b}_{n},\qquad\tilde{b}_{n,E}=\frac{1}{2}\tilde{b}_{n}^{2}.$

Note that, up to now, we have not used the constants in (29b), but found some
general conditions on the their growth as $n$ increases. By using (29b), we
have

$\displaystyle\tilde{a}_{n,E}$
$\displaystyle=\rho\sqrt{1-\rho^{2}}\left(\sqrt{2\log
n}-\frac{1}{2}\frac{\log\left(4\pi\log n\right)}{\sqrt{2\log n}}\right)$
$\displaystyle\tilde{b}_{n,E}$ $\displaystyle=\rho^{2}\log
n-\frac{\rho^{2}}{2}\log\left(4\pi\log
n\right)+\frac{\rho^{2}}{16}\frac{{(\log\left(4\pi\log n\right))}^{2}}{\log
n}.$

Replacing the constants in (43) it yields

$\zeta_{1}=\rho^{2}\left(1+\frac{1}{\rho}\sqrt{\frac{2(1-\rho^{2})}{\log
n}}y-\frac{1}{2}\frac{\log\left(4\pi\log n\right)}{\log n}+o\left({(\log
n)}^{-1}\right)\right).$ (46)

Now, notice that with $\zeta_{1}$ defined in (46), the condition
$\rho^{2}<\min\\{\zeta_{1}(n,y),1\\}$ is always satisfied for large $n$ and
for $\rho\in(0,1)$, as it yields $\rho^{2}<\rho^{2}+\mathcal{O}\left({(\log
n)}^{-1/2}\right)$. Hence, since

$\begin{split}\sqrt{\zeta_{1}}&=\rho{\left(1+\frac{1}{\rho}\sqrt{\frac{2(1-\rho^{2})}{\log
n}}y-\frac{1}{2}\frac{\log\left(4\pi\log n\right)}{\log n}+o\left({(\log
n)}^{-1}\right)\right)}^{1/2}\\\
&\underset{n\to\infty}{\sim}\rho\left(1+\frac{1}{2\rho}\sqrt{\frac{2(1-\rho^{2})}{\log
n}}y-\frac{1}{4}\frac{\log\left(4\pi\log n\right)}{\log
n}-\frac{1-\rho^{2}}{4\log n}y^{2}+o\left({(\log
n)}^{-1}\right)\right),\end{split}$

the expression of $\kappa(\zeta_{1},1)$ becomes

$\kappa(\zeta_{1},1)=1+\frac{\rho^{2}}{2\log n}y^{2}+o\left({(\log
n)}^{-1}\right).$

By using these expressions into Equation (40), we have

$\begin{split}&\mathbb{P}\\!\left(Y_{E}>\zeta_{1}(n,y)\log n\mid X_{E}>\log
n\right)\\\
&\underset{n\to\infty}{\sim}\frac{\exp\left\\{\frac{2\rho^{2}-\rho\left(\sqrt{\zeta_{1}}+1/\sqrt{\zeta_{1}}\right)}{2(1-\rho^{2})}\log\left(4\pi\log
n\right)+\frac{1-\rho/\sqrt{\zeta_{1}}}{2(1-\rho^{2})}\log\zeta_{1}+(1-\kappa(\zeta_{1},1))\log
n\right\\}}{{(1-\rho^{2})}^{-3/2}(\sqrt{\zeta_{1}}-\rho)(1-\rho\sqrt{\zeta_{1}})}.\end{split}$

Since

$\begin{split}\left(1-\frac{\rho}{\sqrt{\zeta_{1}}}\right)\log\zeta_{1}\underset{n\to\infty}{\sim}&\frac{\log\rho}{\rho}\sqrt{\frac{2(1-\rho^{2})}{\log
n}}y+\frac{1}{2\rho^{2}}\frac{2\left(1-\rho^{2}\right)}{\log n}y^{2}\\\
&-\frac{\log\rho}{2}\frac{\log\left(4\pi\log n\right)}{\log
n}-\frac{1-\rho^{2}}{2}\log\rho\frac{y^{2}}{\log n}+o\left({(\log
n)}^{-1}\right)\\\ =&\frac{\log\rho}{\rho}\sqrt{\frac{2(1-\rho^{2})}{\log
n}}y-\frac{\log\rho}{2}\frac{\log\left(4\pi\log n\right)}{\log n}\\\
&+\left(\log\rho-\frac{2}{\rho^{2}}\right)\frac{1-\rho^{2}}{2\log
n}y^{2}+o\left({(\log n)}^{-1}\right),\\\ \end{split}$
$\begin{split}(\sqrt{\zeta_{1}}-\rho)(1-\rho\sqrt{\zeta_{1}})\underset{n\to\infty}{\sim}&\frac{{(1-\rho^{2})}^{3/2}}{\sqrt{2logn}}y-\rho\frac{1-\rho^{2}}{4}\frac{\log\left(4\pi\log
n\right)}{\log n}\\\ &-\rho\frac{1-\rho^{4}}{4\log n}y^{2}+o\left({(\log
n)}^{-1}\right)\end{split}$

and

$\begin{split}\frac{2\rho^{2}-\rho\left(\sqrt{\zeta_{1}}+1/\sqrt{\zeta_{1}}\right)}{2(1-\rho^{2})}\underset{n\to\infty}{\sim}&-\frac{1}{2}+\frac{1}{4\rho}\sqrt{\frac{2(1-\rho^{2})}{\log
n}}y-\frac{1}{8}\frac{\log\left(4\pi\log n\right)}{\log n}\\\
&-\frac{1-\rho^{2}}{8\log n}y^{2}+o\left({(\log n)}^{-1}\right),\end{split}$

where we neglected all the terms characterized by a faster decay with respect
to ${(\log n)}^{-1}$, we obtain

$\mathbb{P}\\!\left(Y_{E}>\zeta_{1}(n,y)\log n\mid X_{E}>\log
n\right)\underset{n\to\infty}{\sim}\frac{1}{y\sqrt{2\pi}}\textrm{e}^{-{(\rho
y)}^{2}/2}.$

∎

###### Proof of Corollary 2.

Notice that

$\mathbb{P}\\!\left(Y_{E}>\zeta_{1}(n,y)\log n\mid X_{E}>\log
n\right)\underset{n\to\infty}{\sim}\frac{\rho}{\rho
y\sqrt{2\pi}}\textrm{e}^{-{(\rho y)}^{2}/2}=\rho\frac{\varphi(\rho y)}{\rho
y},$

where $\varphi$ denotes the probability density function of the standard
normal distribution. Under the assumption that $y$ is large, it is
straightforward to show, by using the Mill’s ratio, that

$\mathbb{P}\\!\left(Y_{E}>\zeta_{1}(n,y)\log n\mid X_{E}>\log
n\right)\underset{n\to\infty}{\sim}\rho\left(1-\Phi(\rho y)\right),$

which completes the proof. ∎

##  Appendix B Appendix - Discussion on $k$

The present section contains some graphical results of the examples considered
in Section 3. We recall that each example runs a simulation of $5000$
replicates of $100000$ bivariate sample $(X,Y)$. As a result, we obtain a
bivariate sample for $(V_{1},V_{2})$ of dimension $5000$ (each replicate of
the $(X,Y)$ sample gives one couple of $(V_{1},V_{2})$), for each $k$. The
values of $k$ indicating the dimension of the extreme set are fixed to
$1,50,100,250,500,1000,1500,2000$. However, the figures do not show the
results for each of the considered values: This choice is made to foster the
readability, as a lot of almost similar plots on a single page may be
redundant. From a quick look at Figures 4 to 7 for both examples, we can draw
the same important conclusions:

  1. (a)

The range of the entries relative to $V_{1}$ is much wider than the range of
the entries corresponding to $V_{2}$;

  2. (b)

The cloud of black points in the surfaces in Figure 4 and Figure 6 becomes
closer and closer to the $v_{1}$ axis as $k$ increases.

This is a clear effect of the positive dependence between $(X,Y)$. To see
this, think about the definition of $(V_{1},V_{2})$: $V_{1}$ is the maximum of
the concomitants of the largest $k$ order statistics and $V_{2}$ is the
maximum of the remaining set. The clear difference between the ranges of the
two samples indicates that the highest concomitants contribute to the values
of $V_{1}$ and the lowest concomitants are responsible of the relatively low
values of $V_{2}$. Moreover, as $k$ increases, almost all the values that
$V_{2}$ takes are lower than $1$, while $V_{1}$ can assume values which are
greater than $100$, for example. Explained differently, high values of
concomitants generally correspond to high order statistics of $X$.

It has to be said that the plotting range of the surfaces is chosen in order
to enhance the visualization purposes, as extremely wide intervals would have
been not suitable to observe the results. However, there can be points which
fall outside the plotting area: Figure LABEL:fig:_SC_surface_k_1000 contains
more points in the middle-right part, while Figure
LABEL:fig:_SC_surface_k_2000 contains more points in the upper-right part.
This indicates that when $k$ is $1000$, some concomitants high in value do not
belong to the extreme set, but when $k$ increases to $2000$, the partition to
which they belong changes. In other words, $V_{1}$ "steals" high values to
$V_{2}$ when $k$ increases.

Plots relative to the marginal distributions support the presented
conclusions: There is no noticeable difference among the log cdf shown in
Figure LABEL:fig:_SC_marginals_V1 and Figure LABEL:fig:_Logistic_marginals_V1,
which means that the sample in $V_{1}$ does not change so much, while the
upper end point of the distribution of $V_{2}$ appears to shrink towards $0$
(Figure LABEL:fig:_SC_marginals_V2 and Figure
LABEL:fig:_Logistic_marginals_V2). This is again an indication of the
"stealing" argument: Since high values of concomitants contributing to $V_{2}$
pass into the extreme set as $k$ increases, only low values of concomitants
are left to compute $V_{2}$, which becomes smaller and smaller. As previously
said, this is the effect of the positive dependence between $(X,Y)$.

Finally, the extremely low errors shown in the boxplots in Figure
LABEL:subfig:_SC_boxplots and Figure LABEL:subfig:_Logistic_boxplots ensure
that the asymptotic approximation is almost perfect. The same conclusion can
be made from the plots sowing the marginal errors in Figure 5 and Figure 7.
Despite some variability for low values of $v_{1}$ and $v_{2}$ (which is small
anyway), the errors rapidly approach $0$ as $v_{1}$ and $v_{2}$ increase,
which is important for practical applications as we are usually interested in
studying what happens over high thresholds.

(a)

(b)  (c)

(d)  (e)  (f)

(g)

Figure 4: Survival Clayton$(\theta=2)$. Bivariate cdf computed by numerically
integrating equation (19), for various values of $k$. The black bullets
correspond to the bivariate empirical cdf of the simulated samples. Panel g)
shows the boxplots of the errors in absolute value between the bivariate
asymptotic cdf computed in $(V_{1,i},V_{2,i})$ for $i=1,\dots,5000$ and the
corresponding empirical bivariate cdf. The gold dots represent the error
means.

(a)

(b)

Figure 5: Survival Clayton$(\theta=2)$. Empirical univariate cdf of $V_{1}$
and $V_{2}$, together with the asymptotic theoretical values obtained by
numerically integrating the asymptotic joint cdf (19).

(a)

(b)  (c)

(d)  (e)  (f)

(g)

Figure 6: Logistic$(\gamma=1/2)$. Bivariate cdf computed by numerically
integrating equation (19), for various values of $k$. The black bullets
correspond to the bivariate empirical cdf of the simulated samples. Panel g)
shows the boxplots of the errors in absolute value between the bivariate
asymptotic cdf computed in $(V_{1,i},V_{2,i})$ for $i=1,\dots,5000$ and the
corresponding empirical bivariate cdf. The gold dots represent the error
means.

(a)

(b)

Figure 7: Logistic$(\gamma=1/2)$ example. Empirical univariate cdf of $V_{1}$
and $V_{2}$, together with the asymptotic theoretical values obtained by
numerically integrating the asymptotic joint cdf (19).

##  Appendix C Appendix - The Von Mises conditions

This section reports the Von Mises conditions, as written in Resnick (1987),
borrowing his notation.

Proposition 1.15 (Resnick (1987)): Suppose $F$ is absolutely continuous with
positive density $f$ in some neighborhood of $\infty$.

  1. ()

If for some $\alpha>0$

$\lim_{x\rightarrow\infty}xf(x)/(1-F(x))=\alpha$ (47)

then $F\in\mathscr{D}\left(\Phi_{\alpha}\right)$. We may choose $a_{n}$ to
satisfy $a_{n}f\left(a_{n}\right)\sim\alpha/n$.

  2. ()

If $f$ is nonincreasing and $F\in\mathscr{D}\left(\Phi_{\alpha}\right)$ then
(47) holds.

  3. ()

Equation (47) holds iff for some $z_{0}$ and all $x>z_{0}$, we have

$1-F(x)=c\exp\left\\{-\int_{z_{0}}^{x}t^{-1}\alpha(t)\,\textrm{d}{t}\right\\}$

where $\lim_{t\rightarrow\infty}\alpha(t)=\alpha$ and $c$ is a positive
constant.

Proposition 1.16 (Resnick (1987)): Suppose $F$ has finite right end-point
$x_{0}$ and is absolutely continuous in a left neighborhood of $x_{0}$ with
positive density $f$.

  1. ()

If for some $\alpha>0$

$\lim_{x\uparrow x_{0}}\left(x_{0}-x\right)f(x)/(1-F(x))=\alpha$ (48)

then $F\in\mathscr{D}\left(\Psi_{\alpha}\right)$.

  2. ()

If $f$ is nonincreasing and $F\in\mathscr{D}\left(\Psi_{\alpha}\right)$ then
(48) holds.

  3. ()

Equation (48) holds iff $c(x)$ can be taken to be constant in some left
neighborhood of $x_{0}$ in the representation

$1-F(x)=c(x)\exp\left\\{-\int_{x_{0}-1}^{x}\delta(t)/\left(x_{0}-t\right)\,\textrm{d}{t}\right\\}\quad\text{
and for }x<x_{0}$

where $\lim_{t\uparrow x_{0}}\delta(t)=\alpha,\,\lim_{t\uparrow
x_{0}}c(t)=c_{0}>0$.

Proposition 1.17 (Resnick (1987)): Let $F$ be absolutely continuous in a left
neighborhood of $x_{0}$ with density $f$.

  1. ()

If

$\lim_{t\uparrow
x_{0}}f(x)\int_{x}^{x_{0}}(1-F(t))\,\textrm{d}{t}/(1-F(x))^{2}=1$ (49)

then $F\in D(\Lambda)$. In this case we may take

$\displaystyle f(t)$
$\displaystyle=\int_{t}^{x_{0}}(1-F(s))\,\textrm{d}{s}/(1-F(x))$
$\displaystyle b_{n}$ $\displaystyle=(1/(1-F))^{\leftarrow}(n),\quad
a_{n}=f\left(b_{n}\right).$

  2. ()

If $f$ is nonincreasing and $F\in\mathscr{D}(\Lambda)$ then holds.

  3. ()

Equation (49) holds iff

$1-F(x)=c\exp\left\\{-\int_{z_{0}}^{x}(g(t)/f(t))\,\textrm{d}{t}\right\\},\quad
z_{0}<x<x_{0},$ (50)

where $\lim_{t\uparrow_{0}}g(x)=1$ and $f$ is absolutely continuous with
density $f(x)\rightarrow 0$ as $x\uparrow x_{0}$.

  4. ()

Equation (49) or (50) are equivalent to
$tf\left((1/(1-F))^{\leftarrow}(t)\right)\in{RV}_{0}$ (Sweeting (1985)).

[Acknowledgments] This paper was mainly written during the first author’s
visiting period at the ESSEC Business School, Paris, France and during the
second author’s visiting period at the ESOMAS Department, University of Turin,
Turin, Italy. Both authors are grateful to the hosting institutions. The
authors are grateful for funding to Labex MME DII (ANR-11-LABX-0023-01) and
ESOMAS department (University of Turin).

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