# Background risk model in presence of heavy tails under dependence

Dimitrios G. Konstantinides, Charalampos D. Passalidis  Dept. of Statistics
and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi,
GR-83 200 Samos, Greece<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the background risk model under a various forms of dependence and
some distribution classes of heavy tails. First, we study the asymptotic
behavior of tail expectation of a portfolios with unequal heavytailedness, of
non-random sums and randomly stopped sums, under some dependence structure
which contains the independence as a special case. Further we investigate the
asymptotic behavior of a pair of randomly weighted sums, generalizing the
dependence structure among random vector components, our results contains the
finite ruin probability in bi-dimensional discrete time risk model with
unequal heavytailedness. Next, we carry out asymptotic analysis of the tail
distortion risk measures in background risk model under various forms of
dependence, with multivariate regularly varying risk distributions in each
portfolio.

Keywords: Multivariate regular variation; Conditional dependence; Breiman’s
theorem; Tail distortion risk measure; Tail expectation; Strongly asymptotic
independence
Mathematics Subject Classification: Primary 62P05; Secondary 91G05; 91B05.

##  1\. Introduction

We have a $n$ business lines insurance and we approximated some risk measures,
in order to evaluate its efficiency over a fixed time interval. Let us
consider a random vector ${\bf X}=(X_{1},\,\ldots,\,X_{n})\in\mathbb{R}^{n}$,
that depicts the losses in the $n$ lines of insurance business, in this
interval. Next, we denote by ${\bf\Theta}=(\Theta_{1},\,\ldots,\,\Theta_{n})$
the vector, describing the systemic risk factors.

Hence, the vector ${\bf\Theta}{\bf
X}=(\Theta_{1}\,X_{1},\,\ldots,\,\Theta_{n}\,X_{n})$ express the present value
of the claims at each business line, while the sum

$\sum_{i=1}^{n}\Theta_{i}\,X_{i}\,,$

denotes the total claim. In most papers on such models, consider that the main
claims $X_{1},\,\ldots,\,X_{n}$ are equally weighted, and usually accept as
framework the class of regular variation. This means that the portfolios are
homogeneous with respect to heavytailedness, see for example [38], [54] and
[10].

However, the homogeneity of the tails can be very restrictive, so we consider
the asymptotic behavior of a new risk measure, under non-equally weighted
portfolios. Furthermore, we take into account that for two portfolios the
number of claims is random, and then we show the asymptotic behavior of this
new risk measure.

Finally, we study the tail distortion risk measure, on homogeneous portfolios
under multivariate regular variation and we find the joint asymptotic behavior
of two randomly weighted sums, under a new dependence structure. This form of
dependence permits dependencies among the members of the sequences of the main
claims but also between the two sequences. In most of these results the
independence is included as special case.

Now we give some preliminaries results and notation. Let denote ${\bf
x}=(x_{1},\,\ldots,\,x_{n})$, the scalar product $c\,{\bf
x}=(c\,x_{1},\,\ldots,\,c\,x_{n})$, $x\wedge y=\min\\{x,\,y\\}$, $x\vee
y=\max\\{x,\,y\\}$, $x+=x\vee 0$, $x-=(-x\vee 0)$ and $e_{i}$ represents the
vector, whose all the components are $0$, except the $i$-th, that is $1$. For
two positive functions $f$ and $g$ we write $f(x)=O(g(x))$, as $x\to\infty$,
if

$\limsup_{x\rightarrow\infty}\frac{f(x)}{g(x)}<\infty\,.$

and $f(x)=o(g(x))$, as $x\to\infty$, if

$\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0\,.$

Let denote by $F(x)=\mathbf{P}(X\leq x)$ the distribution of the random
variable $X$ and by $\overline{F}(x):=1-F(x)=\mathbf{P}(X>x)$, its tail. For
two-dimensional functions all the convergences holds as $x\wedge y\to\infty$,
unless otherwise stated. Next, we consider the value at risk

$\displaystyle VaR_{p}(X):=\inf\\{x\in\mathbb{R}\;:\;F(x)\geq p\\}\,,$

for some $p\in(0,\,1)$ and the corresponding conditional tail expectation

$\displaystyle CTE_{p}(X)=\mathbf{E}[X\;|\;X>VaR_{p}(X)]\,.$

Let see now the classes of heavy-tailed distributions and their properties. We
assume that all the distributions have infinite right endpoint, that means
$\overline{F}(x)>0$ for all $x>0$. We say that distribution $F$ has heavy
tail, and write $\overline{F}\in\mathcal{H}$, if for any $\epsilon>0$ the
relation

$\int_{-\infty}^{\infty}e^{\epsilon x}\overline{F}(dx)=\infty\,,$

is true.

We say that distribution $F$ has long tail (symbolically $F\in\mathcal{L}$) if
for some (or equivalently for all) $a\neq 0$ holds

$\lim_{x\rightarrow\infty}\dfrac{\overline{F}(x+a)}{\overline{F}(x)}=1\,.$

The class $\mathcal{L}$ represents a subclass of heavy-tailed distributions
$\mathcal{L}\subset\mathcal{H}$. If $\overline{F}\in\mathcal{L}$, then there
exists some function $a:[0,\infty)\rightarrow[0,\infty)$ such that
$a(x)\rightarrow\infty$, $a(x)=o(x)$, and $\overline{F}(x\pm
a(x))\sim\overline{F}(x)$, as $x\rightarrow\infty$. This $a(x)$ is called
insensitivity function.

We say that distribution $F$ belongs to the class of subexponential
distributions, and write $\overline{F}\in\mathcal{S}$, if for some (or
equivalently for any) $n=2,\,3,\,\ldots$ holds

$\frac{\overline{F}^{n*}(x)}{\overline{F}(x)}\sim n$

when $x\rightarrow\infty$, where $F^{n*}$ is the n-fold convolution of
distribution $F$, introduced in [18].

We say that $F$ has dominatedly varying tail, and write
$\overline{F}\in\mathcal{D}$, if holds

$\limsup_{x\rightarrow\infty}\dfrac{\overline{F}(t\,x)}{\overline{F}(x)}<\infty$

for some (or equivalently, for all) $0<t<1$. Let us make clear, that neither
$\mathcal{D}\subset\mathcal{H}$ nor $\mathcal{H}\subset\mathcal{D}$ are true.
However
$\mathcal{D}\cap\mathcal{L}=\mathcal{D}\cap\mathcal{S}\subset\mathcal{L}$.
Now, we remind some properties of regular variation. A random variable $X$
with distribution $F$ is regularly varying with index $\alpha>0$ and write
$\overline{F}\in\mathcal{R}_{-\alpha}$ if

$\dfrac{\overline{F}(tx)}{\overline{F}(x)}\rightarrow t^{-\alpha}\,,$

as $x\rightarrow\infty$, uniformly for any $t>0$. The class of distributions
with regularly varying tails is contained in $\mathcal{D}\cap\mathcal{L}$

$\mathcal{R}:=\bigcup_{\alpha>0}\mathcal{R}_{-\alpha}\subset\mathcal{D}\cap\mathcal{L}\subset\mathcal{S}\subset\mathcal{L}\subset\mathcal{H}$

Let us now consider the limits

$\overline{F}_{*}(t):=\liminf_{x\rightarrow\infty}\dfrac{\overline{F}(tx)}{\overline{F}(x)}\,,\qquad\overline{F}^{*}(t):=\limsup_{x\rightarrow\infty}\dfrac{\overline{F}(tx)}{\overline{F}(x)}\,,$

for all $t>0$.

For some distribution $F$ the upper and lower Matuszewska indexes are given by

$\alpha_{F}:=\inf\left\\{-\dfrac{\ln\overline{F}_{*}(t)}{\ln
t}:t>1\right\\}\,,\qquad\beta_{F}:=\sup\left\\{-\dfrac{\ln\overline{F}^{*}(t)}{\ln
t}:t>1\right\\}\,.$

respectively. For these indexes, that appeared in [39], the following
relations hold. $\overline{F}\in\mathcal{D}$ if and only if
$0\leq\beta_{F}\leq\alpha_{F}<\infty$ and
$\overline{F}\in\mathcal{R}_{-\alpha}$ if and only if
$\beta_{F}=\alpha_{F}=\alpha$, (see [5]).

The class of regularly varying distributions have many closure properties, see
for example [30]. On of this properties is the asymptotic behavior of the tail
product convolution, which is the popular Breiman’s Theorem. In the [7], and
[19] established the following result. If $X$ and $\Theta$ two independent
random variables with $X\in\mathcal{R}_{-a}$ for some $\alpha>0$ and $\Theta$
is non-negative such that $\mathbf{E}[\Theta^{\alpha+\epsilon}]<\infty$ for
some $\epsilon>0$ then
$\mathbf{P}(X\,\Theta>x)\sim\mathbf{E}(\Theta^{\alpha})\mathbf{P}(X>x)$, as
$x\rightarrow\infty$, which means that $X\,\Theta\in\mathcal{R}_{-\alpha}$.

Now we can go to the extension of regular variation in random vectors Let
${\bf X}$ be a random vector in the space $[0,\infty]^{n}$. We remind that
${\bf X}$ follows the multivariate regularly varying distribution, if there
exists a function $b\;:\;\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ and a non-
degenerated Radon measure $v$, such that holds in the space
$[0,\infty]^{n}\setminus\\{{\bf 0}\\}$

$t\mathbf{P}\left(\dfrac{{\bf X}}{b(t)}\in B\right)\stackrel{{\scriptstyle
v}}{{\rightarrow}}\nu(B)\,,$ (1.1)

where it refers to the vague convergence, as $t\rightarrow\infty$ and for the
measure $\nu\;:\;\mathbb{R}^{n}\rightarrow\mathbb{R}$ it holds
$\nu((x,\infty])=x^{-\alpha}$, and we write ${\bf X}\in
MRV(\alpha,\,b,\,\nu)$. This measure $\nu$ is homogeneous, whence for any
Borel set $B\subseteq[0,\infty]^{n}\setminus\\{{\bf 0}\\}$ we obtain

$\nu(tB)=t^{-\alpha}\nu(B)\,,$

for any $t\geq 0$.

For the normalizing function $b(\cdot)$ we have that
$b(\cdot)\in\mathcal{R}_{1/\alpha}$, as is indicated in [42]. Another
representation of (1.1) is in the following form

$\dfrac{1}{\overline{F}(t)}\mathbf{P}\left(\dfrac{{\bf X}}{t}\in
B\right)\stackrel{{\scriptstyle v}}{{\rightarrow}}\nu(B)$

as $t\rightarrow\infty$.

######  Remark 1.1.

The components of the random vector ${\bf X}\in MRV(\alpha,\,b,\,\nu)$ have
the same tail convergence speed, in the sense that each component belongs to
$\mathcal{R}_{-\alpha}$ with the same $\alpha>0$, however this does not mean
that the components have the same distribution, as they eventually differ in
the part of the slowly varying functions.

Now let us remind the strong asymptotic independence (see [34, Assumption A]).
Let $X$ and $Y$ two real-valued random variables with distributions $F$ and
$G$ respectively. We say that $X$ and $Y$ are strongly asymptotically
independent (SAI) if

$\displaystyle\mathbf{P}(X^{-}>x,\;Y>y)$ $\displaystyle=$ $\displaystyle
O[F(-x)\,\overline{G}(y)]\,,\quad\mathbf{P}(X>x,\;Y^{-}>y)=O[\overline{F}(x)\,G(-y)]\,,$
$\displaystyle\mathbf{P}(X>x,\;Y>y)$ $\displaystyle\sim$ $\displaystyle
C\,\overline{F}(x)\,\overline{G}(y)\,,$

as $(x,\,y)\rightarrow(\infty,\,\infty)$, for some constant $C>0$.

######  Remark 1.2.

It is easy to see that, SAI contains the independence as a special case. In
the case where $X$ and $Y$ are non-negative (or generally bounded from below)
as in our case, then $X$ and $Y$ are SAI if holds

$\mathbf{P}(X>x,\;Y>y)\sim C\overline{F}(x)\,\overline{G}(y)$

as $(x,\,y)\rightarrow(\infty,\,\infty)$. The SAI covers a wide spectrum of
dependence as for example Ali-Mikhail-Haq, Farlie-Gumbel-Morgenstern and Frank
copulas.

The rest of this paper structure is the following. In Section 2 we study one
double background risk model, more specifically we study the tail expectation
of this model and we give our main result for tail expectation of unequal
heavytailedness portfolios under dependence.In section 3 we extend the jointly
tail behavior of two weighted random sums, given in [33], in the case of
random vectors with generalized tail asymptotically independent components.
This dependence structure enables the study of the dependence between two
random vectors. Additionally we give the ruin probability in bi-dimensional
discrete time risk model under the assumptions of the results of this section.
In Section 4, some extensions Breiman’s theorem are presented which used to
establish asymptotic results of tail distortion risk measure in various
background risk models considering the dependence forms between systemic risk
factors and losses.

##  2\. Tail expectation under unequal heavy-tailedness

###  2.1. Non-Random Sums

In this section we study the tail expectation in the case with portfolios in
MRV but with different tail indexes. We denote

$X_{n}(\Theta):=\sum_{i=1}^{n}\Theta_{i}\,X_{i}\,,\qquad
Y_{m}(\Delta):=\sum_{j=1}^{m}\Delta_{j}\,Y_{j}\,.$

We can consider this model as a double background risk model, which has reason
to be studied as well explain below. We give the tail expectation as follows

$\displaystyle
TE[X_{n}(\Theta),\;Y_{m}(\Delta),\;u]:=\mathbf{E}\left[X_{n}(\Theta)+Y_{m}(\Delta)\;\big{|}\;X_{n}(\Theta)+Y_{m}(\Delta)>u\right]\,,$

for every $n,\,m\in{\mathbb{N}}$ and for some large enough $u>0$.

The components of the vectors $(X_{1},\,\ldots,\,X_{n})$ and
$(Y_{1},\,\ldots,\,Y_{m})$ represent primary, non-degenerated random
variables, depicting the main risks. The components of the
$(\Theta_{1},\,\ldots,\,\Theta_{n})$ and $(\Delta_{1},\,\ldots,\,\Delta_{m})$
represent the systemic risk factors.

This subsection focused to the following question: what happens when we want
to study n-independent portfolios, that are asymptotic marginal distributions,
with tail $\overline{F}\in\mathcal{R}_{-\alpha}$ and other m-independent
portfolios, that are asymptotic marginal distributions, with tail
$\overline{G}\in\mathcal{R}_{-\alpha^{\prime}}$?

  1. (1)

If $\alpha=\alpha^{\prime}$, then we can study the merging of the two random
vectors into one multivariate regular varying vector with $m+n$ components,
that are arbitrarily dependent, because $n\wedge m$ random variables are
dependent with SAI. There is no reason to use the tail expectation, but we
used Corollary 4.2, with $g(x)=x$, and for some $u$ not necessarily equal to
VaR, for the only sum of $m+n$ components.

  2. (2)

But if X and Y are dependent with $\alpha\neq\alpha^{\prime}$ then things
change. So we study the relation of tail expectation, under a certain
dependence, called strong asymptotic independence.

The goal of this subsection is the asymptotic double-bound for
$TE(X_{n}(\Theta),Y_{m}(\Delta))$ under the assumptions at the beginning of
this section. Furthermore this double-bound can be done asymptotic equivalent
under one extra condition.

The next result can be found in [33, Corollary 1].

######  Proposition 2.1 (Li (2018)).

Let ${(X_{i},\,Y_{i}),\;i\in{\mathbb{N}}}$ be a sequence of mutually
independent random vectors with marginal distributions
$\overline{F}_{i}\in\mathcal{R}_{-\alpha_{i}}$ and
$\overline{G}_{i}\in\mathcal{R}_{-\alpha^{\prime}_{i}}$, for some
$\alpha_{i},\;\alpha^{\prime}_{i}\in[0,\infty)$. Assume that for every pair
$(X_{i},\,Y_{i})$ the components are SAI, with constant $C_{i}>0$. Let
${S_{i},\,\Delta_{i},\;i\in{\mathbb{N}}}$ be a sequence of non-negative (and
non-degenerated) arbitrarily dependent random variables such that

$\mathbf{E}\left[\left(\Theta_{i}^{(\alpha_{i}-\delta)+}\vee\Theta_{i}^{(\alpha_{i}+\delta)}\right)\left(\Delta_{j}^{(\alpha^{\prime}_{j}-\delta)+}\vee\Delta_{j}^{(\alpha^{\prime}_{j}+\delta)}\right)\right]<\infty\,,$
(2.1)

for some $\delta>0$ and for any $i,\,j\;\in{\mathbb{N}}$. Assume that
$(X_{i},\,Y_{i}),\;i\in{\mathbb{N}}$ and
${\Theta_{i},\,\Delta_{i},\;i\in{\mathbb{N}}}$ are independent. Then

$\displaystyle\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y)$ (2.2)
$\displaystyle\qquad\sim\sum_{i=1}^{n}\sum_{1=j\neq
i}^{m}\mathbf{E}\left[\Theta_{i}^{\alpha_{i}}\,\Delta_{j}^{\alpha^{\prime}_{j}}\right]\overline{F}_{i}(x)\,\overline{G}_{j}(y)+\sum_{i=1}^{n\wedge
m}C_{i}\,\mathbf{E}\left[\Theta_{i}^{\alpha_{i}}\,\Delta_{i}^{\alpha^{\prime}_{i}}\right]\overline{F}_{i}(x)\,\overline{G}_{i}(y)\,,$

as $(x,\,y)\rightarrow(\infty,\,\infty)$.

######  Remark 2.1.

Because we study multivariate regular variation, the marginal distributions
have the same tail index, i.e. $\overline{F}_{i}\in\mathcal{R}_{-\alpha}$ and
$\overline{G}_{j}\in\mathcal{R}_{-\alpha^{\prime}}$ then (2.1), (2.2) are

$\mathbf{E}\left[\left(\Theta_{i}^{(\alpha-\delta)+}\vee\Theta_{i}^{(\alpha+\delta)}\right)\left(\Delta_{j}^{(\alpha^{\prime}-\delta)+}\vee\Delta_{j}^{(\alpha^{\prime}+\delta)}\right)\right]<\infty\,,$
(2.3)

for some $\delta>0$ and for any $i,j\in{\mathbb{N}}$ and

$\mathbf{P}(X_{n}(\Theta)>x,Y_{m}(\Delta)>y)\sim
K\,\overline{F}(x)\,\overline{G}(y),$ (2.4)

as $(x,\,y)\rightarrow(\infty,\,\infty)$, where we denote

$K:=\sum_{i=1}^{n}\sum_{1=j\neq
i}^{m}\mathbf{E}\left[\Theta_{i}^{\alpha}\,\Delta_{j}^{\alpha^{\prime}}\right]+C\,\sum_{i=1}^{n\wedge
m}\mathbf{E}\left[\Theta_{i}^{\alpha}\,\Delta_{i}^{\alpha^{\prime}}\right]\,.$

In fact, we study one background risk model, with $n$-portfolios that are part
of some n-dimensional regular varying vector with independent components
(therefore the $n$-portfolios are independent) with tail index ”$\alpha$” and
$m$-independent portfolios that are part of $m$-dimensional multivariate
regular varying vector with independent components and with tail index
”$\alpha^{\prime}$”. The systemic risk factors
$\Theta_{1},\,\ldots,\,\Theta_{n}$ are independent and are independent of the
primary components $X_{1},\,\ldots,\,X_{n}$ (similarly for
$\Delta_{1},\,\ldots,\,\Delta_{m}$ and $Y_{1},\,\ldots,\,Y_{m}$). Additionally
the $X_{i}$ and $Y_{i}$ are SAI while $X_{i}$ and $Y_{j}$ are independent for
any $i\neq j$, and $\Theta_{i},\Delta_{i}$ are arbitrarily dependent (for the
same $i$).

######  Theorem 2.1.

Consider two vectors $\textbf{X}\in MRV(\alpha,\,b,\,\nu)$ and $\textbf{Y}\in
MRV(\alpha^{\prime},\,b^{\prime},\,\nu^{\prime})$, whose components are non-
negative and independent. Let us assume that $X_{i}$ and $Y_{i}$ are SAI with
marginal distributions F and G, respectively, where $\alpha>1$ and
$\alpha^{\prime}>1$ and ${\bf\Theta}$ and ${\bf\Delta}$ are non-negative,
arbitrarily dependent random vectors, but with independent components. Assume
that $({\bf\Theta},\;{\bf\Delta})$ and $(\textbf{X},\;\textbf{Y})$ are
independent and the ${\bf\Theta}$, ${\bf\Delta}$ satisfy the relationship
(2.3). Then

$\displaystyle u+K\int_{u}^{\infty}\overline{F}(x)\overline{G}(x)dx\lesssim
TE[X_{n}(\Theta),\;Y_{m}(\Delta),\;u]$ $\displaystyle\lesssim
u+\dfrac{1}{K\overline{F}(u)\overline{G}(u)}\sum_{i=1}^{n}\left[\mathbf{E}[\Theta_{i}^{a}]\int_{u}^{\infty}\overline{F}(x)dx+\sum_{j=1}^{m}\mathbf{E}[\Delta_{j}^{a^{\prime}}]\int_{u}^{\infty}\overline{G}(x)dx\right],$
(2.5)

as $u\rightarrow\infty$.

###### Proof.

We denote

$\overline{F}_{\Theta}(x,\,n):=\mathbf{P}(X_{n}(\Theta)>x)\,,\quad\overline{G}_{\Delta}(x,\,m):=\mathbf{P}(Y_{m}(\Delta)>x)\,.$

. We can write

$\displaystyle
TE[X_{n}(\Theta),\;Y_{m}(\Delta),\;u]=\mathbf{E}[X_{n}(\Theta)+Y_{m}(\Delta)\;|\;X_{n}(\Theta)+Y_{m}(\Delta)>u]$
$\displaystyle=\int_{0}^{\infty}\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>x\;|\;X_{n}(\Theta)+Y_{m}(\Delta)>u)\,dx=$
$\displaystyle\left(\int_{0}^{u}+\int_{u}^{\infty}\right)\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>x\>|\>X_{n}(\Theta)+Y_{m}(\Delta)>u)dx=:J_{1}(u)+J_{2}(u).$

For $J_{1}(u)$ we have

$J_{1}(u)=\int_{0}^{u}\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>x\;|\;X_{n}(\Theta)+Y_{m}(\Delta)>u)dx=\int_{0}^{u}1dx=u\,.$
(2.6)

Now we estimate the lower bound of $J_{2}(u)$

$\displaystyle J_{2}(u)$ $\displaystyle=$
$\displaystyle\int_{u}^{\infty}\dfrac{\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>x)}{\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>u)}dx\,.$
(2.7)

For the lower bound of integral, using relation (2.4), and dominated
convergence theorem (because $\alpha,\alpha^{\prime}>1$) we obtain

$\displaystyle
J_{2}\geq\int_{u}^{\infty}\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>x)\,dx$
$\displaystyle\geq$
$\displaystyle\int_{u}^{\infty}\mathbf{P}(X_{n}(\Theta)>x\,,\;Y_{m}(\Delta)>x)\,dx$
(2.8) $\displaystyle\sim$ $\displaystyle
K\,\int_{u}^{\infty}\overline{F}(x)\,\overline{G}(x)\,dx\,,$

as $u\to\infty$, which give the lower bound. For the denominator, taking into
account that the random variables are non-negative and further because of the
closure property with respect to product convolution (see [7], because the
assumptions of finite moments in (2.3)) and the closure property of the
regular variation with respect to convolution (namely from
$X_{n}(\Theta)\in\mathcal{R}_{-\alpha}$ and
$Y_{m}(\Delta)\in\mathcal{R}_{-\alpha^{\prime}}$), by non-negative assumption
of the components we find out

$\displaystyle\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>x)$ $\displaystyle\leq$
$\displaystyle\mathbf{P}\left(X_{n}(\Theta)>\dfrac{x}{2}\right)+\mathbf{P}\left(Y_{m}(\Delta)>\dfrac{x}{2}\right)$
(2.9) $\displaystyle\sim$ $\displaystyle
2^{\alpha}\,\overline{F}_{\Theta}(x,\,n)+2^{\alpha^{\prime}}\,\overline{G}_{\Delta}(x,\,m)\,.$

as $x\to\infty$. Using again the expression in (2.7) for $J_{2}(u)$, we can
estimate the upper bound as well. Indeed, for the denominator in (2.7), we
have the asymptotic estimation

$\mathbf{P}(X_{n}(\Theta)+Y_{m}(\Delta)>u)\geq\mathbf{P}(X_{n}(\Theta)>u\,,\;Y_{m}(\Delta)>u)\sim
K\,\overline{F}(u)\,\overline{G}(u)\,,$ (2.10)

as $u\to\infty$. Therefore, from (2.9) and (2.10) we find the following upper
bound

$J_{2}(u)\lesssim\dfrac{1}{K\,\overline{F}(u)\,\overline{G}(u)}\,\left(\int_{u}^{\infty}2^{\alpha}\overline{F}_{\Theta}(x,\,n)\,dx+\int_{u}^{\infty}2^{\alpha^{\prime}}\overline{G}_{\Delta}(x,\,m)\,dx\right)\,,$
(2.11)

as $u\to\infty$. Thus from (2.8) and (2.11) we have that :

$\displaystyle K\int_{u}^{\infty}\overline{F}(x)\,\overline{G}(x)\,dx\lesssim
TE[X_{n}(\Theta)\,,\;Y_{m}(\Delta)\,,\;u]-u$
$\displaystyle\qquad\qquad\qquad\lesssim\dfrac{1}{K\,\overline{F}(u)\,\overline{G}(u)}\,\int_{u}^{\infty}\left[2^{\alpha}\overline{F}_{\Theta}(x,\,n)+2^{\alpha^{\prime}}\overline{G}_{\Delta}(x,\,m)\right]\,dx\,,$
(2.12)

as $u\rightarrow\infty$ For the calculation of $\overline{F}_{\Theta}(x,\,n)$
and $\overline{G}_{\Delta}(x,\,m)$, because of independence between
$(\textbf{X},\textbf{Y})$ and $({\bf\Theta},{\bf\Delta})$ we can used
Breiman’s theorem. First, we see if
$\mathbf{P}(X_{i}>u)=\overline{F}(u)\in\mathcal{R}_{-\alpha}$, then we find
$\Theta_{i}\,X_{i}\in\mathcal{R}_{-\alpha}$ and

$\mathbf{P}(\Theta_{i}\,X_{i}>u)\sim\mathbf{E}\left[\Theta_{i}^{\alpha}\right]\,\mathbf{P}(X_{i}>u)=\mathbf{E}\left[\Theta_{i}^{\alpha}\right]\,\overline{F}(u)\,,$

as $u\rightarrow\infty$. If ${\bf\Theta}$ has independent components, then by
Theorem 2.3 of [31], we have:

$\mathbf{P}\left(\sum_{i=1}^{n}\Theta_{i}\,X_{i}>u\right)\sim\sum_{i=1}^{n}\mathbf{P}(\Theta_{i}\,X_{i}>u)\,,$
(2.13)

as $u\rightarrow\infty$, because of
$\Theta_{i}\,X_{i}\in\mathcal{R}_{-\alpha}\subset\mathcal{\mathcal{D}\cap\mathcal{L}}$.
Therefore with similar arguments for $\Delta_{j}\,Y_{j}$ , and via Breiman’s
theorem we have:

$\overline{F}_{\Theta}(u,\,n)\sim\overline{F}(u)\sum_{i=1}^{n}\mathbf{E}\left[\Theta_{i}^{\alpha}\right]\,,\qquad\overline{G}_{\Delta}(u,\,m)\sim\overline{G}(u)\sum_{j=1}^{m}\mathbf{E}\left[\Delta_{j}^{\alpha^{\prime}}\right]\,,$

as $u\rightarrow\infty$. Therefore under the same assumptions we obtain (2.1).
∎

For more details about the asymptotic behavior of

$\mathbf{P}\left(\sum_{i=1}^{n}\Theta_{i}X_{i}>x\right)\,,$

in a variety of dependence structures and heavy tail classes see [45], [46],
[50], [53] among others. We note that the conditions $\alpha>1$ and
$\alpha^{\prime}>1$ are necessary, because without these assumptions the
primary random variables they do not hane first moment, as a result the tail
expectation does not exists in this case.

###  2.2. Randomly stopped sums

Let study now the tail expectation in two randomly stopped sums of the form

$X_{N}(\Theta):=\sum_{i=1}^{N}\Theta_{i}\,X_{i}\,,\qquad
Y_{M}(\Delta):=\sum_{j=1}^{M}\Delta_{j}\,Y_{j}\,.$ (2.14)

where $N$ and $M$ represent positive, integer random variables, with finite
right endpoint. We can again imagine that we consider two portfolios with main
claims $\\{X_{i}\\}$ and $\\{Y_{j}\\}$, where $i,\,j\in{\mathbb{N}}$, while
the $\\{\Theta_{i}\\}$ and $\\{\Delta_{j}\\}$, where $i,\,j\in{\mathbb{N}}$,
represent the discount factors. Namely, the difference from the set up in the
previous subsection is that the number of claim arrivals in each portfolio is
a random variable.

In this subsection we study the tail expectation for randomly stopped sums
presented in (2.14). The risk measures are usually considered for non-random
sums, however the insurance practice needs several times randomly stopped
sums, since in non-life risk models is not known in advanced the number of
claims in a certain time interval. For recent information on risk measures in
renewal risk model see [35]. Motivated by the shortage of these topics in
scientific literature, in combination with increasing interest on stopped
sums, see for example [6], [41], [49], [29], we examine the tail expectation
in two non-equally weighted portfolios. The present set up can not be included
in the MRV one when the regular variation indexes $\alpha$ and
$\alpha^{\prime}$ are equal, because of the random variables $N$ and $M$.
Thus, this case represent interest, in opposite with the non-random sums,
where we mentioned that the case $\alpha=\alpha^{\prime}$ is not interesting
as special case of previous results.

######  Assumption 2.1.

Let $\\{(X_{i},\,Y_{i})\,,\;i\in{\mathbb{N}}\\}$, is a sequence of independent
and identically distributed non-negative random pairs, with the general pair
$(X,\,Y)$ to be strongly asymptotic independent ($SAI$) and where their
distributions $F\in\mathcal{R}_{-\alpha}$ and
$G\in\mathcal{R}_{-\alpha^{\prime}}$ respectively, with
$\alpha,\,\alpha^{\prime}>0$. The
$\\{(\Theta_{i},\,\Delta_{i})\,,\;i\in{\mathbb{N}}\\}$ are non-negative,
independent random pais, non-degenerated to zero, such that

$\sum_{i=1}^{\infty}\mathbf{E}\left[\Theta_{i}^{\alpha+\delta}\right]^{1/\rho}+\sum_{j=1}^{\infty}\mathbf{E}\left[\Delta_{j}^{\alpha^{\prime}+\delta}\right]^{1/\rho}+\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\mathbf{E}\left[\Theta_{i}^{\alpha+\delta}\,\Delta_{j}^{\alpha^{\prime}+\delta}\right]<\infty\,,$
(2.15)

for some $\rho=1+\delta$. Let $N$ and $M$ positive, integer random variables
with finite right endpoint. Further we assume that the $\\{N,\,M\\}$,
$\\{(X_{i},\,Y_{i})\,,\;i\in{\mathbb{N}}\\}$ and
$\\{(\Theta_{i},\,\Delta_{i})\,,\;i\in{\mathbb{N}}\\}$ and mutually
independent

Under Assumption 2.1, the dependence between the two randomly stopped sums
follows from the following three points

  1. (1)

Each pair of discount factors $(\Theta_{i},\,\Delta_{i})$ is arbitrarily
dependent.

  2. (2)

Each pair of main claims $(X_{i},\,Y_{i})$ is strongly asymptotic in dependent
($SAI$).

  3. (3)

The counting random variables $\\{N,\,M\\}$ are arbitrarily independent.

From the fact that $N$ and $M$ have finite right endpoint, we observe that

$\mathbf{E}\left[e^{k\,N}\right]+\mathbf{E}\left[e^{k\,M}\right]<\infty\,,$

for some $k>0$.In the following proposition we provide a restriction of [33,
Cor. 3].

######  Proposition 2.2.

[Li (2018)] Under Assumption 2.1 we obtain

$\mathbf{P}\left[X_{N}(\Theta)>x\,,\;Y_{M}(\Delta)>y\right]\sim
K^{\prime}\,\overline{F}(x)\,\overline{G}(y)\,,$ (2.16)

as $(x,\,y)\to(\infty,\,\infty)$, where we denote

$K^{\prime}:=\mathbf{E}\left[\sum_{i=1}^{N}\sum_{i\neq
j=1}^{M}\mathbf{E}\left[\Theta_{i}^{\alpha}\,\Delta_{j}^{\alpha^{\prime}}\right]+C\,\sum_{i=1}^{N\wedge
M}\mathbf{E}\left[\Theta_{i}^{\alpha}\,\Delta_{i}^{\alpha^{\prime}}\right]\right]\,.$
(2.17)

The following result provides asymptotics bounds for the tail expectation of
randomly stopped sums.

######  Theorem 2.2.

Under Assumption 2.1 we find

$\displaystyle
K^{\prime}\,\int_{u}^{\infty}\overline{F}(x)\,\overline{G}(x)\,dx$
$\displaystyle\lesssim$ $\displaystyle
TE[X_{N}(\Theta)\,,\;Y_{M}(\Delta)\,,\;u]-u$ $\displaystyle\lesssim$
$\displaystyle\dfrac{1}{K^{\prime}\,\overline{F}(u)\,\overline{G}(u)}\,\int_{u}^{\infty}\left[2^{\alpha}\,\overline{F}_{\Theta}(x,\,N)+2^{\alpha^{\prime}}\,\overline{G}_{\Delta}(x,\,M)\right]\,dx\,.$

as $u\to\infty$, where
$\overline{F}_{\Theta}(x,\,N)=\mathbf{P}[X_{N}(\Theta)>x]$ and
$\overline{G}_{\Delta}(x,\,M)=\mathbf{P}[Y_{M}(\Delta)>x]$.

###### Proof.

Initially, as in the proof of Theorem 2.1, we obtain

$\displaystyle
TE[X_{N}(\Theta)\,,\;Y_{M}(\Delta)\,,\;u]=\mathbf{E}[X_{N}(\Theta)+Y_{M}(\Delta)\;|\;X_{N}(\Theta)+Y_{M}(\Delta)>u]$
(2.19)
$\displaystyle=\left(\int_{0}^{u}+\int_{u}^{\infty}\right)\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>x\;|\;X_{N}(\Theta)+Y_{M}(\Delta)>u\right]\,dx=:I_{1}(u)+I_{2}(u)\,.$

Following the argument in relation (2.6), we get

$\displaystyle I_{1}(u)=u\,.$ (2.20)

For the lower bound of $I_{2}(u)$ we find

$\displaystyle I_{2}(u)$ $\displaystyle=$
$\displaystyle\int_{u}^{\infty}\dfrac{\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>x\right]}{\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>u\right]}\,dx\geq\int_{u}^{\infty}\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>x\right]\,dx$
(2.21) $\displaystyle\geq$
$\displaystyle\int_{u}^{\infty}\mathbf{P}\left[X_{N}(\Theta)>x\,,\;Y_{M}(\Delta)>x\right]\,dx\sim
K^{\prime}\,\int_{u}^{\infty}\overline{F}(x)\,\overline{G}(x)\,dx\,,$

as $u\to\infty$, where in the pre-last step we use the fact that
$X_{N}(\Theta)\geq 0$ and $Y_{M}(\Delta)\geq 0$ an Bd the elementary
inequality $\mathbf{P}[A\cap B]\leq\mathbf{P}[A]$. Thence, from (2.21) we have
the lower bound of (2.2).

Now we look for the upper bound of $I_{2}(u)$. At first we obtain

$\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>u\right]\geq\mathbf{P}\left[X_{N}(\Theta)>u\,,\;Y_{M}(\Delta)>u\right]\sim
K^{\prime}\,\overline{F}(u)\,\overline{G}(u)\,,$ (2.22)

as $u\to\infty$. Further from (2.15) we find

$\mathbf{E}\left[\Theta_{i}^{\alpha+\delta}\right]<\infty\,,\;\mathbf{E}\left[\Delta_{j}^{\alpha^{\prime}+\delta}\right]<\infty\,,$

for some $\delta>0$ and for any $i,\,j\in{\mathbb{N}}$. Therefore applying
Breiman’s theorem we have $\Theta_{i}\,X_{i}\in\mathcal{R}_{-\alpha}$ and
$\Delta_{j}\,Y_{j}\in\mathcal{R}_{-\alpha^{\prime}}$, for any
$i,\,j\in{\mathbb{N}}$. But since $\Theta_{i}\,X_{i}$ is independent of
$Theta_{k}\,X_{k}$ for any $k\neq i$ and $\Delta_{j}\,Y_{j}$ is independent of
$\Delta_{l}\,Y_{l}$ for any $l\neq j$ from Assumption 2.1, and the $N$ and $M$
have finite right endpoint, by [44, Th. 5] we get that the distributions of
$X_{N}(\Theta)$ and $Y_{M}(\Delta)$ belong to the classes
$\mathcal{R}_{-\alpha}$ and $\mathcal{R}_{-\alpha^{\prime}}$ respectively.
Hence,

$\displaystyle\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>x\right]$
$\displaystyle\leq$
$\displaystyle\mathbf{P}\left[X_{N}(\Theta)>\dfrac{x}{2}\right]+\mathbf{P}\left[Y_{M}(\Delta)>\dfrac{x}{2}\right]$
(2.23) $\displaystyle\sim$ $\displaystyle
2^{\alpha}\,\overline{F}_{\Theta}(x,\,N)+2^{\alpha^{\prime}}\,\overline{G}_{\Delta}(x,\,M)\,,$

as $x\to\infty$. So, from relations (2.22) and (2.23) follows

$\displaystyle I_{2}(u)$ $\displaystyle=$
$\displaystyle\int_{u}^{\infty}\dfrac{\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>x\right]}{\mathbf{P}\left[X_{N}(\Theta)+Y_{M}(\Delta)>u\right]}\,dx$
(2.24) $\displaystyle\lesssim$
$\displaystyle\dfrac{1}{K^{\prime}\,\overline{F}(u)\,\overline{G}(u)}\int_{u}^{\infty}\left[2^{\alpha}\,\overline{F}_{\Theta}(x,\,N)+2^{\alpha^{\prime}}\,\overline{G}_{\Delta}(x,\,M)\right]\,dx\,,$

as $x\to\infty$. This way, from relations (2.19), (2.20) and (2.24) we
conclude also the upper bound of (2.2). ∎

######  Remark 2.2.

If we want to proceed in analysis of the distributions $F_{\Theta}(x,\,N)$ and
$G_{\Delta}(x,\,M)$, then by [44, Th. 5] in combination with the last theorem,
we obtain the asymptotic behavior of the two tails

$\displaystyle\overline{F}_{\Theta}(x,\,N)$ $\displaystyle\sim$ $\displaystyle
x^{-\alpha}\sum_{n=1}^{\check{N}}\mathbf{P}\left[N=n\right]\,\sum_{k=1}^{\check{N}}L_{k}(x)\,,$
$\displaystyle\overline{G}_{\Delta}(x,\,M)$ $\displaystyle\sim$ $\displaystyle
x^{-\alpha^{\prime}}\sum_{m=1}^{\check{M}}\mathbf{P}\left[M=m\right]\,\sum_{l=1}^{\check{M}}L^{\prime}_{l}(x)\,,$
(2.25)

as $x\to\infty$, where $\check{N}$ and $\check{M}$ denote the right endpoints
of $N$ and $M$ respectively and the functions $L_{k}$ and $L^{\prime}_{l}$ are
slowly varying and stemming from the representations

$\displaystyle\mathbf{P}[\Theta_{k}\,X_{k}>x]$ $\displaystyle=$ $\displaystyle
x^{-\alpha}\,L_{k}(x)\,,$ $\displaystyle\mathbf{P}[\Delta_{l}\,Y_{l}>x]$
$\displaystyle=$ $\displaystyle x^{-\alpha^{\prime}}\,L^{\prime}_{l}(x)\,.$
(2.26)

Thence, relation (2.2) through (2.2) renders

$\displaystyle
K^{\prime}\int_{u}^{\infty}\overline{F}(x)\,\overline{G}(x)\,dx$
$\displaystyle\lesssim$ $\displaystyle
TE[X_{N}(\Theta)\,,\;Y_{M}(\Delta)\,,\;u]-u$ $\displaystyle\lesssim$
$\displaystyle\dfrac{1}{K^{\prime}\,\overline{F}(u)\,\overline{G}(u)}\Bigg{[}\left(2^{\alpha}\,\sum_{n=1}^{\check{N}}\mathbf{P}\left[N=n\right]\right)\int_{u}^{\infty}x^{-\alpha}\,\sum_{k=1}^{\check{N}}L_{k}(x)\,dx$
$\displaystyle\qquad\qquad+\left(2^{\alpha^{\prime}}\,\sum_{m=1}^{\check{M}}\mathbf{P}\left[M=m\right]\right)\int_{u}^{\infty}x^{-\alpha^{\prime}}\,\sum_{l=1}^{\check{M}}L^{\prime}_{l}(x)\,dx\Bigg{]}\,,$

as $u\to\infty$.

##  3\.  Generalized tail asymptotic independence

Next we establish the following relation

$\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y)\sim\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\,,$
(3.1)

as $x\wedge y\rightarrow\infty$, with
$\Theta_{i},\,\Delta_{j},\;i\in{\mathbb{N}}$ arbitrarily dependent non-
negative random variables and the primary risk components
$\\{(X_{i},\,Y_{i}),\;i\in{\mathbb{N}}\\}$ independent random vectors, while
$X_{i}$ and $Y_{i}$ are SAI (but $X_{i}$ and $Y_{j}$ are independent for any
$i\neq j$),
$\mathbf{P}(X_{i}>x)=\overline{F}_{i}(x)\in\mathcal{D}\cap\mathcal{L}$ and
$\mathbf{P}(Y_{j}>x)=\overline{G}_{j}(x)\in\mathcal{D}\cap\mathcal{L}$. With
joint behavior of two randomly weighted sums see also [12], [43].

Although several dependence structures on finite sequences of random variables
have been studied in recent years (see [13], [25], [37]), the case of two
sequences appears rarely. The dependence structure between different sequences
has good reason to be studied. In the next definition we examine triples of
random variables, where probably there exist different dependence structure in
each pair. For example in the triple $(X_{i},\,X_{k},\,Y_{j})$, the pair
$(X_{i},\,X_{j})$ are arbitrarily dependent and the $(X_{i},\,Y_{j})$ are SAI.

######  Definition 3.1.

Let $X_{1},\,\ldots,\,X_{n}$ and $Y_{1},\,\ldots,\,Y_{m}$ real valued random
variables. Then we say that $X_{1},\,\ldots,\,X_{n},\,Y_{1},\,\ldots,\,Y_{m}$
are generalized tail asymptotic independent, symbolically GTAI, if hold the
following relations

$\lim_{\min({x_{i},x_{k},y_{j})}\rightarrow\infty}\mathbf{P}(|X_{i}|>x_{i}\;\mid\;X_{k}>x_{k},\;Y_{j}>y_{j})=0\,,$
(3.2)

for all $1\leq k\notin\\{i,\,j\\}\leq n$ and

$\lim_{\min({x_{i},y_{j},y_{k})}\rightarrow\infty}\mathbf{P}(|Y_{j}|>y_{j}\;\mid\;X_{i}>x_{i},\;Y_{k}>y_{k})=0\,,$
(3.3)

for all $1\leq k\notin\\{i,\,j\\}\leq m$.

######  Remark 3.1.

This dependence structure allows dependence between $X_{1},\,\ldots,\,X_{n}$
and dependence between $X_{i}$ and $Y_{j}$ (not only for $i=j$) and similarly
for $Y_{1},\,\ldots,\,Y_{m}$. In this paper we restrict ourselves in the case
with $X_{i},\,Y_{i}$ to be SAI dependent for the same $i$, and $X_{i},\,Y_{j}$
independent for any $i\neq j$.

Notice that if $X_{i}$ and $Y_{j}$ are independent for any
$i,j\in{\mathbb{N}}$ (i.e. the two sequences are independent) then the
relationships (3.2), (3.3) can be written as follows

$\lim_{x_{i}\wedge
x_{k}\rightarrow\infty}\mathbf{P}(|X_{i}|>x_{i}\;\mid\;X_{k}>x_{k})=0$ (3.4)

holds for all $1\leq i\neq k\leq n$ and

$\lim_{y_{j}\wedge
y_{k}\rightarrow\infty}\mathbf{P}(|Y_{j}|>y_{j}\;\mid\;Y_{k}>y_{k})=0$ (3.5)

for all $1\leq j\neq k\leq m$ Through (3.4), (3.5) we get the definition of
tail asymptotic independence of $X_{1},\,\ldots,\,X_{n}$ and
$Y_{1},\,\ldots,\,Y_{m}$ respectively, introduced in [25].

######  Assumption 3.1.

We assume that the following random variables
$X_{1},\,\ldots,\,X_{n},\,Y_{1},\,\ldots,\,Y_{m}$, are $GTAI$, and the random
discount factors

$\Theta_{1}\,,\ldots,\,\Theta_{n},\,\Delta_{1},\,\ldots,\,\Delta_{m}\,,$

are non-negative and non-degenerated to zero random variables, that are upper
bounded random variables and independent of
$X_{1},\,\ldots,X_{n},\,Y_{1},\,\ldots,\,Y_{m}$.

######  Lemma 3.1.

Under the Assumption 3.1, we have that the products

$\Theta_{1}X_{1},\,\ldots,\,\Theta_{n}X_{n},\,\Delta_{1}Y_{1},\,\ldots,\,\Delta_{m}Y_{m}\,,$

are $GTAI$.

###### Proof.

By definition of $GTAI$, for any $\epsilon>0$, there exist some constant
$\kappa_{0}>0$ such that for any $x_{i},x_{k},y_{j}>\kappa_{0}$ the relation:

$\mathbf{P}(|X_{i}|>x_{i}\mid X_{k}>x_{k},Y_{j}>y_{j})<\epsilon$

holds for all $1\leq k\notin\\{i,j\\}\leq n$. Let

$\mathbf{P}(\Theta_{i}\leq b_{i})=1\,,\qquad\mathbf{P}(\Delta_{j}\leq
d_{j})=1\,,$

where $b_{i},\,d_{j}>0$, for all $i=1,...,n$ and $j=1,...,m$. Then for
$x_{i},x_{k},y_{j}$ sufficiently large, namely

$\dfrac{x_{i}}{b_{i}}>\kappa_{0}\,,\quad\dfrac{x_{k}}{b_{k}}>\kappa_{0}\,,\quad\dfrac{y_{j}}{\lambda_{j}}>\kappa_{0}\,,$

we have that:

$\displaystyle\mathbf{P}(|\Theta_{i}X_{i}|>x_{i},\Theta_{k}X_{k}>x_{k},\Delta_{j}Y_{j}>y_{j})$
$\displaystyle=\int_{0}^{b_{i}}\int_{0}^{b_{k}}\int_{0}^{d_{j}}\mathbf{P}\left(|X_{i}|>\frac{x_{i}}{c_{i}},X_{k}>\frac{x_{k}}{c_{k}},Y_{j}>\frac{y_{j}}{\lambda_{j}}\right)\mathbf{P}(\Theta_{i}\in
dc_{i},\Theta_{k}\in dc_{k},\Delta_{j}\in d\lambda_{j})$
$\displaystyle=\int_{0}^{b_{i}}\int_{0}^{b_{k}}\int_{0}^{d_{j}}\mathbf{P}\left(|X_{i}|>\frac{x_{i}}{c_{i}}\mid
X_{k}>\frac{x_{k}}{c_{k}},Y_{j}>\frac{y_{j}}{\lambda_{j}}\right)\mathbf{P}\left(X_{k}>\frac{x_{k}}{c_{k}},Y_{j}>\frac{y_{j}}{\lambda_{j}}\right)$
$\displaystyle\qquad\times\mathbf{P}(\Theta_{i}\in dc_{i},\Theta_{k}\in
dc_{k},\Delta_{j}\in
d\lambda_{j})\leq\epsilon\mathbf{P}(\Theta_{k}X_{k}>x_{k},\Delta_{j}Y_{j}>y_{j})$

by arbitrariness of $\epsilon>0$, we have the first relation. The symmetrical
relation, namely:

$\mathbf{P}(|\Delta_{j}Y_{j}|>y_{j},\Theta_{i}X_{i}>x_{i},\Delta_{k}Y_{k}>y_{k})\leq\epsilon\mathbf{P}(\Theta_{i}X_{i}>x_{i},\Delta_{k}Y_{k}>y_{k})$

for some $1\leq k\notin\\{i,j\\}\leq m$, can be obtain easily with similar
arguments. And this complete the proof. ∎

######  Assumption 3.2.

Let $\\{(X_{i},\,Y_{i}),\;i\in{\mathbb{N}}\\}$ be some sequence of random
vectors with marginal distributions $F_{i}\in\mathcal{D}\cap\mathcal{L}$ and
$G_{i}\in\mathcal{D}\cap\mathcal{L}$ for all $i\in{\mathbb{N}}$. Assume that
$X_{i}$ and $Y_{i}$ are SAI for the same i and $X_{i},\,Y_{j}$ are independent
for $i\neq j$.

Let $\\{(\Theta_{i},\,\Delta_{i}),\;i\in{\mathbb{N}}\\}$ be some sequence of
arbitrarily dependent, non-negative, non-degenerated and upper bounded random
vectors, such that

$\mathbf{E}\left[\left(\Theta_{i}^{(\beta_{F_{i}}-\delta)+}\vee\Theta_{i}^{\alpha_{F_{i}}+\delta}\right)\,\left(\Delta_{j}^{(\beta_{G_{j}}-\delta)+}\vee\Delta_{j}^{\alpha_{G_{j}}+\delta}\right)\right]<\infty$

for some $\delta>0$ and for any $i,j\in{\mathbb{N}}$. Assume also that
$\\{(X_{i},\,Y_{i}),\;i\in{\mathbb{N}}\\}$ and
$\\{\Theta_{i},\;\Delta_{i},\;i\in{\mathbb{N}}\\}$ are independent.

Next, we need two lemmas for the proof of Theorem 3.1 below.

######  Lemma 3.2.

Under Assumption 3.1 and Assumption 3.2, we find

$\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y,\;\Theta_{k}\,|X_{k}|>a(x))=o\left(\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\right)\,,$

as $x\wedge y\rightarrow\infty$, for any $i,\,j,\,k$ with $i\neq k$, with
$a(x)>0$ and $a(x)\rightarrow\infty$, as $x\rightarrow\infty$, $a(x)<x/2$ and
$a\in\mathcal{R}_{0}$.

###### Proof.

We can see that

$\displaystyle\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y,\;\Theta_{k}\,|X_{k}|>a(x))$
$\displaystyle=\mathbf{P}(\Theta_{k}\,|X_{k}|>a(x)\;\mid\;\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\,\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)$
$\displaystyle=o\left[\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\right]\,,$

as $x\wedge y\rightarrow\infty$, where in the last step we used Assumption
3.1. ∎

Before presenting next lemma, we need some comments. Let suppose that $X$
follows distribution $F$ and $Y$ follows distribution $G$, such that

$\mathbf{P}(X>x,\;Y>y)=O\left(\overline{F}(x)\,\overline{G}(y)\right)\,,$
(3.6)

as $x\wedge y\rightarrow\infty$, whence we can see that there exists some
constant $\Lambda>1$, such that

$\mathbf{P}(X>x,\;Y>y)\leq\Lambda\overline{F}(x)\,\overline{G}(y)\,,$

for any $x,\,y\in\mathbb{R}$. By (3.6) we have that, there exist some
$x_{0}>0$ and $\Lambda_{1}>1$ such that

$\dfrac{\mathbf{P}(X>x,Y>y)}{\overline{F}(x)\,\overline{G}(y)}\leq\Lambda_{1}\,,$

holds for any $x\geq x_{0}$ and for any $y\geq x_{0}$.

We consider two independent random variables $X^{*},\,Y^{*}$ with respectively
tails

$\overline{F^{*}}(x)=1\wedge\left(\sqrt{\Lambda}\>\overline{F}(x)\right)\,,\qquad\overline{G^{*}}(y)=1\wedge\left(\sqrt{\Lambda}\>\overline{G}(y)\right)\,,$

thence we see

$\overline{F^{*}}(x)\sim\sqrt{\Lambda}\>\overline{F}(x)\,,\qquad\overline{G^{*}}(y)\sim\sqrt{\Lambda}\>\overline{G}(y)\,,$
(3.7)

as $x\rightarrow\infty$ and $y\rightarrow\infty$ respectively. Thus for any
$x,\,y\in\mathbb{R}$

$\displaystyle\overline{F}(x)\leq\overline{F^{*}}(x)\,,\quad\overline{G}(y)\leq\overline{G^{*}}(y)\,,$
$\displaystyle\mathbf{P}[X>x,\,Y>y]\leq\mathbf{P}[X^{*}>x,\,Y^{*}>y]\,,$

Now we introduce sequences $\\{X^{\circ}_{i},i\in{\mathbb{N}}\\}$ and
$\\{Y^{\circ}_{i},i\in{\mathbb{N}}\\}$. These sequences have i.i.d. components
and are independent each other. Their distribution tails are

$\overline{F^{\circ}}(x)=1\wedge\left(\sqrt{\Lambda}\>\overline{F^{*}}(x)\right)\,,\qquad\overline{G^{\circ}}(y)=1\wedge\left(\sqrt{\Lambda}\>\overline{G^{*}}(y)\right)\,.$

Therefore

$\overline{F^{\circ}}(x)\sim\sqrt{\Lambda}\>\overline{F^{*}}(x)\,,\qquad\overline{F^{\circ}}\sim\Lambda\overline{F}(x)\,,$
(3.8)

as $x\rightarrow\infty$ and

$\overline{G^{\circ}}(y)\sim\sqrt{\Lambda}\>\overline{G^{*}}(y)\sim\Lambda\,\overline{G}(y)\,,$
(3.9)

as $y\rightarrow\infty$. Further, for any $x,\,y\in\mathbb{R}$ and for any
$(i,\,j)\in{\mathbb{N}}^{2}$

$\displaystyle\overline{F}_{i}(x)\leq\overline{F^{*}}_{i}(x)\leq\overline{F^{\circ}}_{i}(x)\,,$
$\displaystyle\overline{G}_{j}(y)\leq\overline{G^{*}}_{j}(y)\leq\overline{G^{\circ}}_{j}(y)$
$\displaystyle\mathbf{P}(X_{i}>x,Y_{j}>y)\leq\mathbf{P}(X^{*}_{i}>x,Y^{*}_{j}>y)\leq\mathbf{P}(X^{\circ}_{i}>x,Y^{\circ}_{j}>y)\,.$

######  Lemma 3.3.

Let $\\{(X_{i},\,Y_{i}),\;i\in{\mathbb{N}}\\}$ be a sequence of identically
distributed random vectors with corresponding marginal distributions F and G
such that

$\overline{F}\in\mathcal{D}\,,\qquad\overline{G}\in\mathcal{D}\,,$

and (3.6) holds. Let $\\{\Theta_{i},\,\Delta_{i},\;i\in{\mathbb{N}}\\}$ be a
sequence of non-negative and non-degenerated random variables such that

$E_{ij}:=\mathbf{E}[(\Theta_{i}^{(\beta_{F}-\delta)+}\vee\Theta_{i}^{\alpha_{F}+\delta})\,(\Delta_{j}^{(\beta_{G}-\delta)+}\vee\Delta_{j}^{\alpha_{G}+\delta})]<\infty\,,$
(3.10)

for some $\delta>0$ and for any $i,\,j\;\in{\mathbb{N}}$. Assume that

$\\{(X_{i},\,Y_{i}),\;i\in{\mathbb{N}}\\}\,,\qquad\\{(\Theta_{i},\,\Delta_{i}),\;i\in{\mathbb{N}}\\}\,,$

are independent. Then

(i) There exist some $x_{0}>0$ and some $\Lambda>1$ such that for any
$x>x_{0}$ and for any $y>x_{0}$ holds

$\sup_{(i,\,j)\in{\mathbb{N}}^{2}}\dfrac{\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)}{E_{ij}\overline{F}(x)\,\overline{G}(y)}\leq\Lambda$
(3.11)

(ii) If (3.10) holds with both $\Theta_{i}^{(\beta_{F}-\delta)+}$,
$\Delta_{j}^{(\beta_{G}-\delta)+}$, replaced by $1$, then (3.11) holds for any
$x>0$ and any $y>0$.

###### Proof.

For the relations (3.7) and (3.8) we find

$\overline{F}(x)\leq\overline{F^{*}}(x)\leq\sqrt{\Lambda_{1}}\>\overline{F}(x)\,,\quad\overline{F^{*}}(x)\leq\overline{F^{\circ}}(x)\leq\sqrt{\Lambda_{1}}\>\overline{F^{*}}(x)\,,$

therefore we obtain

$\overline{F}(x)\leq\overline{F^{\circ}}(x)\leq\Lambda_{1}\overline{F}(x)\,,\qquad\overline{G}(x)\leq\overline{G^{\circ}}(x)\leq\Lambda_{1}\overline{G}(x)\,,$

for some $\Lambda_{1}>1$ and for any $x\in\mathbb{R}$. For the relations (3.8)
adn (3.9) we have

$\overline{F^{\circ}}\in\mathcal{D}\,,\qquad\overline{G^{\circ}}\in\mathcal{D}\,.$

Next, for the proof of ($i$), ($ii$) we follow [33, Lemma 2 (i), (ii)], with
$X^{*}$ and $Y^{*}$ replaced by $X^{\circ}$ and $Y^{\circ}$ respectively. ∎

######  Theorem 3.1.

Under Assumption 3.1 and Assumption 3.2, for every pair
$(n,\,m)\in{\mathbb{N}}^{2}$ we obtain (3.1).

###### Proof.

We consider the events

$\displaystyle H_{x}^{\pm}:=\left\\{\bigvee_{i=1}^{n}\Theta_{i}X_{i}>x\pm
a(x)\right\\}\,,\qquad
H_{y}^{\pm}:=\left\\{\bigvee_{j=1}^{m}\Delta_{j}Y_{j}>y\pm a(y)\right\\}\,.$

with $a(x)$ defined in Lemma 3.2. Now, let us define the probabilities

$\displaystyle I_{1}(x,\,y)$ $\displaystyle:=$
$\displaystyle\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;H_{x}^{-},\;H_{y}^{-})\,,$
$\displaystyle I_{2}(x,\,y)$ $\displaystyle:=$
$\displaystyle\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;(H_{x}^{-})^{c})\,,$
$\displaystyle I_{3}(x,\,y)$ $\displaystyle:=$
$\displaystyle\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;(H_{y}^{-})^{c})\,,$

whence we can see that

$\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y)\leq
I_{1}(x,\,y)+I_{2}(x,\,y)+I_{3}(x,\,y)\,,$ (3.12)

so we have to estimate the $I_{1}(x,\,y),\,I_{2}(x,\,y),\,I_{3}(x,\,y)$,

$\displaystyle I_{1}(x,\,y)$ $\displaystyle\leq$
$\displaystyle\mathbf{P}(H_{x}^{-},H_{y}^{-})=\mathbf{P}\left(\bigvee_{i=1}^{n}\Theta_{i}\,X_{i}>x-a(x),\;\bigvee_{j=1}^{m}\Delta_{j}\,Y_{j}>y-a(y)\right)$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}\left(\Theta_{i}\,X_{i}>x-a(x),\;\Delta_{j}\,Y_{j}>y-a(y)\right)$
$\displaystyle\sim$
$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}X_{i}>x,\,\Delta_{j}Y_{j}>y)\,,$

as $x\wedge y\rightarrow\infty$, where in the last step we used [33, Lemma
3(ii)] (as far as $X_{i}$ and $Y_{i}$ are SAI). Next

$\displaystyle
I_{2}(x,\,y)=\mathbf{P}\left(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;\bigvee_{i=1}^{n}\Theta_{i}\,X_{i}>\dfrac{x}{n},\;\bigvee_{j=1}^{m}\Delta_{j}\,Y_{j}>\dfrac{y}{m},\;(H_{x}^{-})^{c}\right)$
$\displaystyle=\mathbf{P}\left(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;\bigvee_{i=1}^{n}\Theta_{i}\,X_{i}>\dfrac{x}{n},\;\bigvee_{j=1}^{m}\Delta_{j}\,Y_{j}>\dfrac{y}{m},\;\bigvee_{i=1}^{n}\Theta_{i}\,X_{i}<x-a(x)\right)$
$\displaystyle\leq\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{1=k\neq
i}^{n}\mathbf{P}\left(\Theta_{i}\,X_{i}>\dfrac{x}{m},\;\Delta_{j}\,Y_{j}>\dfrac{y}{m},\;\Theta_{k}\,X_{k}>\dfrac{a(x)}{n}\right)$
$\displaystyle=o\left(\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\right)\,,$

as $x\wedge y\rightarrow\infty$, where in the last step we used Lemma 3.2 and
[33, Lemma 3(i)].

Symmetrically we find

$I_{3}(x,\,y)=o\left(\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\right)\,,$

as $x\wedge y\rightarrow\infty$. So from (3.12) we obtain

$\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y)\lesssim\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\,,$

as $x\wedge y\rightarrow\infty$.

For the lower bound of

$\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y)\,,$

we get the inequality

$\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y)\geq\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;H_{x}^{+},\;H_{y}^{+})\,.$

Applying Bonferroni inequality twice we obtain

$\displaystyle\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;H_{x}^{+},\;H_{y}^{+})$
(3.13)
$\displaystyle=\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(X_{n}(\Theta)>x,\;Y_{m}(\Delta)>y,\;\Theta_{i}X_{i}>x+a(x),\;\Delta_{j}Y_{j}>y+a(y))$
$\displaystyle\quad-\sum_{1\leq i<k\leq
n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}X_{i}>x+a(x),\;\Theta_{k}X_{k}>x+a(x),\;\Delta_{j}Y_{j}>y+a(y))$
$\displaystyle\quad-\sum_{i=1}^{n}\sum_{1\leq j<k\leq
m}\mathbf{P}(\Theta_{i}X_{i}>x+a(x),\;\Delta_{j}Y_{j}>y+a(y),\;\Delta_{k}Y_{k}>y+a(y))\,,$

and by [33, Lemma 2 (iii) and Lemma 3 (i)] the last two terms in (3.13) are
asymptotically negligible with respect to

$\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\,.$

For the first term of right hand side in (3.13) we find a lower bound

$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}X_{i}>x+a(x),\;\Delta_{j}Y_{j}>y+a(y))$
$\displaystyle\quad-\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{1=k\neq
i}^{n}\mathbf{P}\left(\Theta_{i}X_{i}>x+a(x),\;\Delta_{j}Y_{j}>y+a(y),\;\Theta_{k}X_{k}<-\dfrac{a(x)}{n}\right)$
$\displaystyle\quad-\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{1=k\neq
i}^{m}\mathbf{P}\left(\Theta_{i}X_{i}>x+a(x),\;\Delta_{j}Y_{j}>y+a(y),\;\Delta_{k}Y_{k}<-\dfrac{a(x)}{n}\right)\,,$

which by Lemma 3.2 is asymptotically greater than

$\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{P}(\Theta_{i}X_{i}>x,\;\Delta_{j}Y_{j}>y)\,,$

as $x\wedge y\rightarrow\infty$, and thus we have the asymptotic relation
(3.1). ∎

######  Remark 3.2.

We proved asymptotic estimation (3.1), when
$X_{1},\,\ldots,\,X_{n},\,Y_{1},\,\ldots,\,Y_{m}$ are GTAI, as it comes by
Definition 3.1, the $X_{i}$, $Y_{i}$ are SAI and the $\Theta_{i},\;\Delta_{i}$
are upper bounded and non-negative random variables. This restriction is
relatively small compared to the extension made in terms of dependence and at
the same time reasonable in these models, since $\Theta_{i}$ and $\Delta_{j}$
depicts the systemic risk factors.

######  Remark 3.3.

We see that [33, Lemma 1, Lemma 2(iii), Lemma 3] continue to apply under the
generalization of conditions in Theorem 3.1 (with the help of Lemma 3.3).

Now applying [33, Lemma 3 (iii)] in (3.1) and we have the following
consequence.

######  Corollary 3.1.

Under Assumption 3.1 and Assumption 3.2 and if
$F_{i}\in\mathcal{R}_{-\alpha_{i}}$, for any $i\in{\mathbb{N}}$ and
$G_{j}\in\mathcal{R}_{-\alpha^{\prime}_{j}}$, for any $j\in{\mathbb{N}}$, with
$\alpha_{i},\;\alpha^{\prime}_{j}\in[0,\,\infty)$ and if $X_{i},\,Y_{i}$ are
SAI dependent, with some constant $C_{i}>0$, then we obtain (2.2).

In the next corollary we show the asymptotic behavior for the tail
expectation, under conditions of Theorem 3.1, and specifically under
conditions of Corollary 3.1. We have now one extension of Theorem 2.1 in the
case when $X_{1},\,\ldots,\,X_{n},Y_{1},\,\ldots,\,Y_{m}$ are GTAI, in the
case when $X_{i},\,Y_{i}$ are SAI and in the case when $X_{i},\,Y_{j}$ are
independent for any $i\neq j$, but the systemic risk factors
$\Theta_{i},\,\Delta_{i}$ are positive and upper bounded, for all
$i\in{\mathbb{N}}$. Furthermore, because of the grouping into two categories
of regular variation, from relation (2.2) we pass to relation (2.4).

######  Remark 3.4.

We write

$X^{\pm}_{n}(\Theta):=\sum_{i=1}^{n}\Theta_{i}X^{\pm}_{i}\,,\qquad
Y^{\pm}_{m}(\Delta):=\sum_{j=1}^{m}\Delta_{j}Y^{\pm}_{j}\,.$

We see that, the corresponding maximums for $X^{\pm}_{n}(\Theta)$ and
$Y^{\pm}_{m}(\Delta)$, namely $\bigvee_{i=1}^{n}X_{i}(\Theta)$ and
$\bigvee_{j=1}^{m}Y_{j}(\Delta)$, satisfy the inequalities

$\displaystyle\mathbf{P}(X_{n}(\Theta)>x,\,Y_{m}(\Delta)>y)$
$\displaystyle\leq$
$\displaystyle\mathbf{P}\left(\bigvee_{i=1}^{n}X_{i}(\Theta)>x,\;\bigvee_{j=1}^{m}Y_{j}(\Delta)>y\right)$
$\displaystyle\leq$
$\displaystyle\mathbf{P}(X^{+}_{n}(\Theta)>x,\;Y^{+}_{m}(\Delta)>y)\,,$

Therefore Theorem 3.1 and Corollary 3.1 are also satisfied after replacement
with

$\left(\bigvee_{i=1}^{n}X_{i}(\Theta),\;\bigvee_{j=1}^{m}Y_{j}(\Delta)\right)\,.$

Finally, we present one bi-dimensional discrete time risk model, in two (or
four) sub-models with the same stochastic process of surplus.

Recently, the discrete time one-dimensional models have attracted attention by
researchers, see [36], [49], [51]. On the other hand more researchers study
the multivariate risk models, because it is rarely for an insurance company to
operate with one line of business, see [14], [15], [16], [27] among others.

We limit us in only two portfolios and discrete time, where $X_{i}$ and
$Y_{i}$ depicts the net loss in $i$-th period, in the first and second
portfolio respectively, $\Theta_{i}$ and $\Delta_{j}$ they continue to be
systemic risk factors (or discount factors).

Therefore the stochastic discounted value of aggregate insurance losses to
time $n\in{\mathbb{N}}$ describe

$\textbf{S}_{n}:=(S_{1n},\;S_{2n})=\left(x-\sum_{i=1}^{n}\Theta_{i}\,X_{i},\;y-\sum_{j=1}^{n}\Delta_{j}\,Y_{j}\right)\,,$

thence one type of ruin probability is given as

$\psi(x,y,n):=\mathbf{P}\left(\bigvee_{i=1}^{n}X_{i}(\Theta)>x,\;\bigvee_{j=1}^{n}Y_{j}(\Delta)>y\right)\,,$

for any $n\in{\mathbb{N}}$, where $x,y$ are initial capitals in each
portfolio. This ruin probability depicts the probability that both portfolios
have been with negative surplus, but not necessarily simultaneously.

######  Corollary 3.2.

(i) If Assumption 3.1 and Assumption 3.2 are true, then we obtain

$\psi(x,y,n)\sim\sum_{i=1}^{n}\sum_{j=1}^{n}\mathbf{P}(\Theta_{i}\,X_{i}>x,\;\Delta_{j}\,Y_{j}>y)\,,$
(3.14)

as $x\wedge y\rightarrow\infty$.

(ii) Under the conditions of Corollary 3.1 we obtain

$\displaystyle\psi(x,y,n)$ $\displaystyle\sim$
$\displaystyle\mathbf{P}(X_{n}(\Theta)>x,\;Y_{n}(\Delta)>y)$
$\displaystyle\sim$ $\displaystyle\sum_{i=1}^{n}\sum_{1=j\neq
i}^{n}\mathbf{E}\left[\Theta_{i}^{\alpha_{i}}\Delta_{j}^{\alpha^{\prime}_{j}}\right]\overline{F}_{i}(x)\overline{G}_{j}(y)+\sum_{i=1}^{n}C_{i}\mathbf{E}\left[\Theta_{i}^{\alpha_{i}}\Delta_{i}^{\alpha^{\prime}_{i}}\right]\,\overline{F}_{i}(x)\,\overline{G}_{i}(y)\,,$

as $x\wedge y\rightarrow\infty$.

###### Proof.

Directly from Remark 3.4, the definition of ruin probability, Theorem 4.1 and
Corollary 3.1. ∎

######  Remark 3.5.

It is obvious that the net losses of each portfolio do not necessarily have
the same distribution in each period, but must be to the same class (i.e
$\mathbf{P}(X_{i}>x)\in\mathcal{D}\cap\mathcal{L}$ or
$\mathbf{P}(X_{i}>x)\in\mathcal{R}$ for any $i=1,\,\ldots,\,n$) and similarly
for $Y_{j}$. Additionally, relations (3.14) and (3.2) hold under the
assumptions of Theorem 1 and Corollary 1 in [33] respectively.

##  4\. Tail distortion risk measures

Let us define function
$\Pi\,:\,\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}\rightarrow\mathbb{R}_{+}^{n}$,
as follows $\Pi({\bf r},\,{\bf u}):={\bf r}\,{\bf
u}=(r_{1}\,u_{1},\,\ldots,\,r_{d}\,u_{n})$. Then the vector
$\Pi({\bf\Theta},\,{\bf X})={\bf\Theta}\,{\bf
X}=(\Theta_{1}\,X_{1},\,\ldots,\,\Theta_{n}\,X_{n})$, represents the real loss
of the portfolio over the given time interval.

Here limit our study in simple background risk model (without unequal
heavytaildness) and for this reason we have a general class of risk measures
and various forms of dependence between systemic risk factors and losses. In
order to make the model more realistic, we allocate the initial capital into
$d$ lines of business, in general of different amounts. Thus we need non-
random, positive weights $w_{1},\,\ldots,\,w_{n}$, with

$\displaystyle\sum_{i=1}^{n}w_{i}=1\,.$ (4.1)

Then the loss portfolio is presented as

$\displaystyle{\bf\Theta\,X(w)}:=\sum_{i=1}^{n}w_{i}\,\Theta_{i}\,X_{i}\,.$

This model called background risk model and is a special case of systemic risk
model. For more details about Background risk model see [3], [20] and [47].

The class of regular variation although represents a small class of heavy-
tailed distributions, has been studied much deeper than any other class of
heavy-tailed distributions. It comes by the mathematical convenience of its
properties, its frequent appearance in real data in the form of Pareto
distribution, and its easy extension in higher dimensions.

However the most significant feature of MRV is the fact that its components
can be arbitrarily dependent, and have some very good properties. For example,
if ${\bf X}\in MRV$, with index $\mathcal{\alpha}$ then every (non-degenerated
to zero) linear combination of the components of ${\bf X}$ belongs to the
class $\mathcal{R}_{-\alpha}$, see section 7.3 in [42]. As a result, for
$w_{i}>0$, for $i=1,\,\ldots,\,n$, satisfying a normalization condition (4.1)
we find that the weighted sum has distribution with regularly varying tail

$\mathbf{P}\left(\sum_{i=1}^{n}w_{i}\,X_{i}>x\right)\in\mathcal{R}_{-\alpha}\,,$
(4.2)

with the same parameter $\alpha>0$.

In [11] was examined the distortion tail risk measure in a similar background
risk model. In [11] are developed asymptotic results for the tail risk measure
of the quantity ${\bf\Theta\,X(w)}$, where
${\bf\Theta\,X(w)}_{i}=\Theta\,X_{i}$ and the weights $w_{i}$ for
$i=1,\,\ldots,\,n$ are defined as above, with $\Theta$ to be one-dimensional,
and therefore is common for all business lines, and independent of ${\bf X}$,
and also we have continuous distortion function.

In the last section we develop the background risk model, in case of a
systemic risk factor ${\bf\Theta}$, which represents a random vector, that
means the risk factor is not necessary the same in all the businesses, and
then we study several cases, where the ${\bf\Theta}$ and ${\bf X}$ are not
independent.

In our methodology we use the closure property of MRV under the product
convolution, or under the scalar product. Next we remind the extension of
Breiman’s theorem in case of two independent random vectors, one of them
multivariate regularly varying, established in [4].

######  Proposition 4.1 (Basrak-Davis-Mikosch (2002)).

Let two non-negative random vectors ${\bf X}\in MRV(\alpha,\,b,\,\nu)$ and
${\bf\Theta}$ with arbitrarily dependent components and

$\mathbf{E}\left(\Theta_{i}^{\alpha+\varepsilon}\right)<\infty\,,$ (4.3)

for any $i=1,\,\ldots,\,n$ and some $\varepsilon>0$. If the ${\bf\Theta}$ and
${\bf X}$ are independent, then ${\bf\Theta\,X}\in
MRV(\alpha,\,b,\,\nu_{\bf\Theta})$ where

$\nu_{\bf\Theta}(B)=\mathbf{E}\left[\nu\left({\bf\Theta}^{-1}B\right)\right]=\int_{[0,\infty]^{d}}\nu({\bf
s}^{-1}B)\,\mathbf{P}_{\bf\Theta}\,(d{\bf u})\,,$

with $\mathbf{P}_{\bf\Theta}(\cdot)$ a probability measure, corresponding to
distribution of ${\bf\Theta}$.

In this result the independence assumption between the ${\bf X}$ and
${\bf\Theta}$ is definitely restrictive. Therefore in [24] are found the
following three extensions of this result. We present two of them, restricted
only on the positive quadrant. The first assumes from one side ${\bf X}\in
MRV(\alpha,\,b,\,\nu)$ and from the other side that the vectors
${\bf\Theta},\,{\bf X}$ are asymptotically independent (wich contains the
independence assumption as a special case). The asymptotic independence was
introduced in [40].

######  Proposition 4.2 (Fougeres-Mercadier (2012)).

Let ${\bf X}\in MRV(\alpha,\,b,\,\nu)$ and ${\bf\Theta}$ be two non-negative
random vectors and the ${\bf\Theta}$ and ${\bf X}$ be asymptotically
independent, in the sense

$t\,\mathbf{P}\left(\left(\dfrac{\bf X}{b(t)},\;{\bf\Theta}\right)\in
B\right)\stackrel{{\scriptstyle v}}{{\rightarrow}}(\nu\times L)(B)\,,$ (4.4)

as $t\rightarrow\infty$, on the Borel sets
$B\subseteq[0,\infty]^{n}\setminus\\{{\bf 0}\\}\times[0,\infty]^{n}$, $\nu$ is
a Radon measure in $[0,\infty]^{n}\setminus{\bf 0}$ and $L$ as probability
measure on $[0,\infty]^{n}$, corresponds to the distribution of ${\bf S}$. If
for some $\delta>0$ and for any $i=1,\,\ldots,\,n$ hold

$\displaystyle\lim_{\epsilon\rightarrow
0}\limsup_{t\rightarrow\infty}t\,\mathbf{E}\left[\left(\dfrac{\|{\bf
X}\|\,|\Theta_{i}|}{b(t)}\right)^{\delta}\,{\bf 1}_{\\{\|{\bf
X}\|/b(t)<\epsilon\\})}\right]=0\,,$ (4.5)
$\displaystyle\int_{[0,\infty]^{n}}\|{\bf u}\|^{\alpha}\,L(d{\bf
u})<\infty\,,$ (4.6)

with $\|\cdot\|$ any norm in $\mathbb{R}^{n}$ and $\nu_{L}$ is a measure of
the form

$\nu_{L}(A):=(\nu\times L)\\{({\bf r},\,{\bf u})\,,\;{\bf r}\in{\bf
u}^{-1}A\\}=\int_{[0,\infty]^{n}}\nu\left({\bf u}^{-1}\,A\right)\,L(d{\bf
u})\,,$

for any $A\subset[0,\,\infty]^{n}\setminus\\{{\bf 0}\\}$, then
${\bf\Theta\,X}\in MRV(\alpha,\,b,\,\nu_{L})$.

The next result comes from [24] considers the case of ${\bf\Theta}$ and ${\bf
X}$ jointly multivariate regularly varying, and at least one couple of
components are asymptotically dependent, namely there exist normalizing
functions $b(\cdot)$, $c(\cdot)$, such that holds

$t\,\mathbf{P}\left(\left(\dfrac{X_{i}}{b(t)},\;\dfrac{\Theta_{i}}{c(t)}\right)_{i=1}^{n}\in
B\right)\stackrel{{\scriptstyle v}}{{\rightarrow}}\nu_{{\bf\Theta},\,{\bf
X}}(B)\,,$ (4.7)

as $t\rightarrow\infty$, in some non-zero Radon measure on the space
$[0,\,\infty]^{2n}\setminus\\{{\bf 0}\\}$. We assume that the measure
$\nu_{{\bf\Theta},\,{\bf X}}\circ\Pi^{\leftarrow}(\cdot)$ is not degenerated
to zero. Hence, there exists an $i$, such that the couple
$(X_{i},\,\Theta_{i})$ is dependent, namely there exists
$i\in\\{1,\,\ldots,\,n\\}$ such that

$\nu_{{\bf\Theta},\,{\bf X}}^{(i)}((0,\,\infty]^{2})>0\,,$ (4.8)

where $\nu_{{\bf\Theta},\,{\bf X}}^{(i)}(\cdot)$ is the restriction of $\nu$
to its $i$-th components of ${\bf\Theta}$ and ${\bf X}$. The normalizing
functions $b,\;c$ are regularly varying with parameters $1/\kappa$ and
$1/\lambda$ respectively, with $\kappa,\;\lambda>0$.

######  Proposition 4.3 (Fougeres-Mercadier (2012)).

Let hold the relations (4.7) and (4.8), then

${\bf\Theta\,X}\in
MRV\left(\dfrac{\kappa\,\lambda}{\kappa+\lambda},\,b\,c,\,\nu_{{\bf\Theta},\,{\bf
X}}\circ\Pi^{\leftarrow}\right)\,$

Now in the classic one-dimensional Background risk model we study the
asymptotic behavior of tail distortion risk measure which is more general than
conditional tail expectation. The following class of measures was introduced
in [48]. For a given non-decreasing function $g:[0,1]\rightarrow[0,1]$, such
that $g(0)=0$, $g(1)=1$, and for any non-negative random variable $X$, with
distribution $F$, the measure

$\rho_{g}[X]=\int_{0}^{\infty}g\left[\overline{F}(x)\right]\,dx$

is called distorted risk measure and the function $g$ is called distortion
function. It is well-known that the $VaR$ and $CTE$ are distortion risk
measures (see for example [55]).

######  Definition 4.1 (Zhu-Li (2012)).

Let $g\;:\;[0,1]\rightarrow[0,1]$ be non-decreasing, such that $g(0)=0$ and
$g(1)=1$, then the tail distortion risk measure of a non-negative random
variable $X$ with distribution $F$ is given by

$\rho_{g}[X\;|\;X>VaR_{p}(X)]=\int_{0}^{\infty}g\left[\overline{F}_{X|X>VaR_{p}(X)}(x)\right]\,dx$

where $\overline{F}_{X|X>y}(x)=\mathbf{P}(X>x\;|\;X>y)$.

Obviously, the tail distortion risk measure is a distortion risk measure.
Furthermore, if the distortion function is the identical function $g(x)=x$,
then the tail distortion risk measure coincides with the conditional tail
expectation.

From [55] we find that if $X\in\mathcal{R}_{-\alpha}$, with $\alpha>0$ and
additionally

$\int_{1}^{\infty}g\left(\dfrac{1}{y^{\alpha-k}}\right)\,dy<\infty\,,$

for some $0<k<\alpha$, then $\rho_{g}[X|X>VaR_{p}(X)]\sim
C_{\alpha}(g)\,VaR_{p}(X)$, as $p\rightarrow 1$, where

$C_{\alpha}(g):=\int_{0}^{1}y^{-1/\alpha}\,g(dy)=1+\int_{1}^{\infty}g\left(\dfrac{1}{y^{\alpha}}\right)\,dy\,.$

This last result was shown for any distortion function, without continuity
requirement.

From [22] we find for the function $B_{X}$, defined as quantile of
$1/\overline{F}$,

$B_{X}(s):=\left(\dfrac{1}{\overline{F}}\right)^{\leftarrow}(s)=F^{\leftarrow}\left(1-\dfrac{1}{s}\right)\,,$
(4.9)

for any $s\geq 1$, and for some continuous distortion function $g$ the
representation

$\rho_{g}[X\;|\;X>VaR_{p}(X)]=\int_{0}^{1}B_{X}\,\left(\dfrac{1}{y(1-p)}\right)\,dg(y)$
(4.10)

for any non-negative random variable $X$ with distribution $F$.

It is clear that the previous asymptotic expressions in section 4 depend on
the distortion function only though the constant $C_{\alpha}(g)$, and
therefore the quantiles of random variables remain clear of distortion. This
helps in practical applications, since the only we need for the application of
the model, when the distortion function is varying for the same risks, is the
calculation of the integral.

Next, we study a first sub-model of background risk, where ${\bf X}\in
MRV(\alpha,\,b,\,\nu)$, with its components to be eventually dependent and the
systemic risk factor ${\bf\Theta}$ to be asymptotically independent of ${\bf
X}$. So, in this model we permit dependence among the risks of the
$d$-business and dependence among the systemic risk factors with the risks,
but if some of the risks becomes too large, practically if it cross some high
enough threshold, then this dependence between ${\bf\Theta}$ and ${\bf X}$ get
eliminated.

######  Theorem 4.1.

Consider the product ${\bf\Theta\,X}$, with ${\bf\Theta}$ and ${\bf X}$ two
non-negative random vectors. Under the conditions of Proposition 4.2 if (4.3)
holds and

$\int_{1}^{\infty}g\left(\dfrac{1}{y^{\alpha/(1+\zeta)}}\right)dy<\infty\,,$
(4.11)

for some $\zeta>0$ hold, then

$\displaystyle\rho_{g}\left[{\bf\Theta\,X(w)}\;\big{|}\;{\bf\Theta\,X(w)}>VaR_{p}\left({\bf\Theta\,X(w)}\right)\right]\sim
C_{\alpha}(g)\,\dfrac{\gamma_{\bf
w}^{1/\alpha}}{\Gamma_{\alpha}}\,\sum_{i=1}^{n}VaR_{p}\left(\Theta_{i}X_{i}\right)\,,$
(4.12)

as $p\rightarrow 1$, with

$\gamma_{\bf
w}:=\lim_{t\rightarrow\infty}\dfrac{\mathbf{P}({\bf\Theta\,X(w)}>t)}{\mathbf{P}\left(\sum_{i=1}^{n}\Theta_{i}X_{i}>t\right)}\,,\qquad\Gamma_{\alpha}:=\sum_{i=1}^{n}\gamma_{e_{i}}^{1/\alpha}\,.$
(4.13)

###### Proof.

From Proposition 4.2, since relations (4.4), (4.5) and (4.6) are satisfied, we
obtain that ${\bf\Theta\,X}\in MRV(\alpha,\,b,\,\nu_{L})$, therefore
${\bf\Theta\,X(w)}\in\mathcal{R}_{-\alpha}$, see for example (4.2). From
relation (4.10) and the fact that

$VaR_{p}\left({\bf\Theta\,X(w)}\right)=B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)\,,$
(4.14)

we can show

$\dfrac{1}{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)}\int_{0}^{1}B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y(1-p)}\right)\,dg(y)\sim
C_{\alpha}(g)\,,$ (4.15)

as $p\rightarrow 1$. Indeed, from the [23], [21, Th. B.2.18] we get that there
exists a $0<\tilde{p}<1$, which depends on $\zeta>0$, such that for
$\tilde{p}\leq p<1$ and $0<y<1$ holds

$\displaystyle\left|\dfrac{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y(1-p)}\right)-B_{{\bf\Theta\,X(w)}}\,\left(\dfrac{1}{1-p}\right)}{\dfrac{1}{\alpha}\,B_{{\bf\Theta\,X(w)}}\,\left(\dfrac{1}{1-p}\right)}-\dfrac{y^{-1/\alpha}-1}{1/\alpha}\right|$
$\displaystyle\qquad\qquad=\left|\dfrac{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y(1-p)}\right)}{\dfrac{1}{\alpha}\,B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)}-\dfrac{y^{-1/\alpha}}{1/\alpha}\right|\leq
y^{-(1+\zeta)/\alpha}\,.$ (4.16)

Further we have

$\left|\dfrac{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y(1-p)}\right)}{\dfrac{1}{\alpha}\,B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)}\right|-\dfrac{y^{-1/\alpha}}{1/\alpha}\leq\left|\dfrac{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y(1-p)}\right)}{\dfrac{1}{\alpha}\,B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)}-\dfrac{y^{-1/\alpha}}{1/\alpha}\right|\,,$
(4.17)

Hence from relations (4.16) and (4.17) we obtain

$\left|\dfrac{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y\,(1-p)}\right)}{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)}\right|\leq
y^{-1/\alpha}+\dfrac{y^{-(1+\zeta)/\alpha}}{\alpha}\,.$ (4.18)

Since the integral in (4.11) converges for some $\zeta>0$, it follows

$\int_{0}^{1}y^{-1/\alpha}\,g(dy)\leq\int_{0}^{1}y^{-(1+\zeta)/\alpha}\,g(dy)<\infty\,,$

whence we obtain

$\int_{0}^{1}\left(y^{-1/\alpha}+\dfrac{y^{-(1+\zeta)/\alpha}}{\alpha}\right)\,g(dy)<\infty\,.$

Therefore from (4.18) and by dominated convergence theorem, we find

$\displaystyle\lim_{p\rightarrow
1}\int_{0}^{1}\dfrac{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y\,(1-p)}\right)}{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)}\,g(dy)$
$\displaystyle=$ $\displaystyle\int_{0}^{1}\lim_{p\rightarrow
1}\dfrac{B_{{\bf\Theta\,X(w)}}\,\left(\dfrac{1}{y\,(1-p)}\right)}{B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{1-p}\right)}\,g(dy)$
$\displaystyle=$
$\displaystyle\int_{0}^{1}y^{-1/\alpha}\,g(dy)=C_{\alpha}(g)\,,$

where the pre-last equality follows from
$B_{{\bf\Theta\,X(w)}}(\cdot)\in\mathcal{R}_{1/\alpha}$, see (4.9) and [42,
sec. 2.4]. Therefore, relation (4.15) is true.

Furthermore, taking into account relation (4.13) we obtain

$\lim_{p\rightarrow
1}\dfrac{\sum_{i=1}^{n}VaR_{p}(\Theta_{i}X_{i})}{VaR_{p}\left({\bf\Theta\,X(w)}\right)}=\dfrac{\Gamma_{\alpha}}{\gamma_{\bf
w}^{1/\alpha}}\,.$ (4.19)

However we also obtain

$\displaystyle\rho_{g}\left[{\bf\Theta\,X(w)}\;\big{|}\;{\bf\Theta\,X(w)}>VaR_{p}\left({\bf\Theta\,X(w)}\right)\right]=\int_{0}^{1}B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y\,(1-p)}\right)\,g(dy)\,,$

while by (4.14) and (4.15) we find

$\int_{0}^{1}B_{{\bf\Theta\,X(w)}}\left(\dfrac{1}{y\,(1-p)}\right)\,g(dy)\sim
C_{\alpha}(g)\,VaR_{p}\left({\bf\Theta\,X(w)}\right)\,,$

as $p\rightarrow 1$ and consequently from relation (4.19) and the last two
expressions we find relation (4.12). ∎

In the following background risk model, we have ${\bf\Theta}$, ${\bf X}$ with
both dependence and asymptotic dependence, as well as the jointly multivariate
regular variation.

######  Corollary 4.1.

If the assumptions of Proposition 4.3 are true, where ${\bf\Theta}$ and ${\bf
X}$ are non-negative random vectors with property (4.3) and the integral in
(4.11) converges for some $\zeta>0$, then we obtain (4.12) with

$a:=\dfrac{\kappa\,\lambda}{\kappa+\lambda}\,,$ (4.20)

instead of $\alpha$.

###### Proof.

From Proposition 4.3 we get

${\bf X}\in
MRV\left(\dfrac{\kappa\lambda}{\kappa+\lambda},\,b\,c,\,\nu_{{\bf\Theta},\,{\bf
X}}\circ\Pi^{\leftarrow}\right)\,.$

Hence, since $\mathbf{P}(X_{i}\leq x)=F(x)$ with
$\overline{F}\in\mathcal{R}_{-a}$ with $a$ given in (4.20), we obtain
${\bf\Theta\,X(w)}\in\mathcal{R}_{-a}$ and further the proof follows the proof
of Theorem 4.1. ∎

Next we provide a simple corollary, which is an extension of the result from
[11, Theorem 3.1] in case when the systemic risk factor is independent of the
risk vector ${\bf X}\in MRV$.

######  Corollary 4.2.

Let assume that the assumptions of Proposition 4.1 hold, with ${\bf\Theta}$
and ${\bf X}$ non-negative random vectors, that are mutually independent. If
the integral in (4.11) converges for some $\zeta>0$, then we obtain

$\displaystyle\rho_{g}\left[{\bf\Theta\,X(w)}\;\big{|}\;{\bf\Theta\,X(w)}>VaR_{p}\left({\bf\Theta\,X(w)}\right)\right]\sim
C_{\alpha}(g)\,\dfrac{\gamma_{\bf
w}^{1/\alpha}}{\Gamma_{\alpha}}\sum_{i=1}^{n}\left[\mathbf{E}\left(\Theta_{i}^{\alpha}\right)\right]^{1/\alpha}\,VaR_{p}(X_{i})\,,$
(4.21)

as $p\rightarrow 1$.

###### Proof.

Firstly we have ${\bf\Theta}\,{\bf X}\in MRV(\alpha,\,b,\,\nu_{{\bf\Theta}})$.
Hence, we find that ${\bf\Theta\,X(w)}\in\mathcal{R}_{-\alpha}$, from the
assumptions of Proposition 4.1, as a result we obtain (4.12), because
$\Theta_{i}$ and $X_{i}$ are independent (namely a special cace of asymptotic
independence). From [9] we find

$VaR_{p}(X_{i})=VaR_{p}(\Theta_{i}\,X_{i})\sim\left[\mathbf{E}\left(\Theta_{i}^{\alpha}\right)\right]^{1/\alpha}\,VaR_{p}(X_{i})\,,$
(4.22)

as $p\rightarrow 1$. Hence from relation (4.22) and relation (4.12) we obtain
(4.21). ∎

In our effort to enrich the models, we study a slightly different dependence
structure between ${\bf\Theta}$ and X which is called conditional dependence.
This last result in tail distortion risk measure goes back to the case when
the systemic risk factor $\Theta$ is one-dimensional, that means it is common
of all $X_{1},\,\ldots,\,X_{n}$.

We consider $\Theta$ to be one-dimensional, namely common for all components,
and take the restriction to one concrete dependence model, the conditional
dependence model, that covers a wide spectrum of dependence, as for example
the Farlie-Gumbel-Morgenstern copulas or the Ali-Mikhail-Haq copulas.

Let check this form of dependence for $n=1$, in the form of $X$, with
$\mathbf{P}(X\leq x)=F_{1}(x)$, whence we obtain

$\mathbf{P}(X>x\;|\;\Theta=t)\sim h_{1}(t)\overline{F}_{1}(x)\,,$ (4.23)

as $x\to\infty$, with $h_{1}(\cdot):[0,\,\infty)\rightarrow(0,\,\infty)$
measurable function. This dependence model was suggested in [2]. The first
application of conditional dependence in risk models was done in [1]. This
dependence structure has the property to begin independence asymptotically (as
the Asymptotic independence).

In [32] there exists an important extension of this dependence structure.
Especially it was considered a $n$-variate function
$V\;:\;[0,\infty]^{n}\setminus\\{{\bf 0}\\}\rightarrow(0,\infty)$ and a uni-
variate function $h_{2}\;:\;(0,\infty)\rightarrow(0,\infty)$, that satisfies
the inequalities

$0<\inf_{t\in(0,\infty)}h_{2}(t)\leq\sup_{t\in(0,\infty)}h_{2}(t)<\infty\,,$

and let $F_{2}$ be a distribution defined on the interval $(0,\infty)$ with
infinite upper bound, then for any ${\bf y}\in[0,\infty]^{n}\setminus\\{{\bf
0}\\}$ holds

$\mathbf{P}({\bf X}>{\bf y}x\;|\;\Theta=t)\sim V({\bf
y})h_{2}(t)\overline{F}_{2}(x)$ (4.24)

as $x\to\infty$. If $\overline{F}_{2}\in\mathcal{R}_{-\alpha}$, for some
$\alpha>0$ then ${\bf X}\in MRV$, with the same index $\alpha$ as it appears
from [32, remark 2.1].

In [17] was shown that under dependence structure (4.24) between $\Theta$ and
X, the probability of joint excess of the product $\Theta\,\textbf{X}$ comes
in the form

$\mathbf{P}[\Theta\,\textbf{X}>\textbf{y}\,x]=\mathbf{P}[\Theta\,X_{1}>y_{1}\,x,\,\ldots,\,\Theta\,X_{n}>y_{n}\,x]\sim\mathbf{E}[h_{2}(\Theta)]\,V(\textbf{y})\,\mathbf{E}[(\Theta^{*})^{\alpha}]\,\overline{F}_{2}(x)\,,$
(4.25)

as $x\to\infty$, for any
$\textbf{y}\in[0,\,\infty]^{n}\setminus\\{\textbf{0}\\}$, under the condition

$\mathbf{E}[\Theta^{\alpha+\varepsilon}]<\infty\,,$ (4.26)

for some $\varepsilon>0$, where by $\Theta^{*}$ we denote a random variable
independent of random vector X, with distribution

$\mathbf{P}[\Theta^{*}\in
d\theta]=\dfrac{1}{\mathbf{E}[h_{2}(\Theta)]}h_{2}(\theta)\,\mathbf{P}[\Theta\in
d\theta]\,,$ (4.27)

Thence, from relation (4.25), it follows that the vector
$\Theta\,\textbf{X}\in MRV$ with the same variation index $\alpha$. So we find
the next result, appearing in vague form in [17].

######  Proposition 4.4 (Cheng-Konstantinides-Wang (2024)).

If $\textbf{X}\in MRV(\alpha,\,b,\,\nu)$ and $\Theta$ is a non-degenerated to
zero random variable, such that (4.26) is true for some $\varepsilon>0$, then
$\Theta\,\textbf{X}\in MRV$, with the same index $\alpha$.

######  Remark 4.1.

It worth to observe that

  1. (1)

From the dependence structure in relation (4.24), we obtain

$\mathbf{P}[X_{i}>x\;|\;\Theta=\theta]\sim\dfrac{1}{\mathbf{E}[h_{2}(\Theta)]}h_{2}(\theta)\,\overline{F}_{i}(x)\,,$
(4.28)

as $x\to\infty$, for any $i=1,\,\ldots,\,n$, see for example in [32, Rem.
2.1]. Relation (4.28), in fact renders the conditional dependence in (4.23)
for any pair $(X_{i},\,\Theta)$.

  2. (2)

In [28, Rem. 5.3] can be found a comment about the conditions under which the
dependence structure (4.24) contains the independence, as a special case.
Next, we examine the asymptotic independence case between $\Theta$ and X, to
show that Proposition 4.4 does not represent a special case of [24, Prop.
4.2]. If $\Theta$ and X are asymptotically independent, then by relation
(4.24) we find

$\dfrac{\mathbf{P}[\Theta\,X_{1}>y_{1}\,x,\,\ldots,\,\Theta\,X_{n}>y_{n}\,x,\,\Theta=\theta]}{\mathbf{P}[\Theta=\theta]}\sim
V(\textbf{y})\,h_{2}(\theta)\,\overline{F}_{2}(x)\,,$

from where follows

$\mathbf{P}[X_{1}>y_{1}\,x,\,\ldots,\,X_{n}>y_{n}\,x]\sim
V(\textbf{y})\,h_{2}(\theta)\,\overline{F}_{2}(x)\,,$ (4.29)

as $x\to\infty$, where in the second step we use that $\Theta$ and X are
asymptotically independent.

Now, for the random vector X, we say that it has asymptotically dependent
components if

$\lim_{x\to\infty}\dfrac{\mathbf{P}[X_{1}>y_{1}\,x,\,\ldots,\,X_{n}>y_{n}\,x]}{\overline{F}(x)}>0\,,$
(4.30)

while in opposite case we say that it has asymptotically independent
components.

Thus, in case (4.30) is true, there exists some number $\varepsilon>0$, which
represents the limit of the ratio. Therefore by (4.29) we obtain

$\varepsilon\,\overline{F}_{2}(x)\sim
V(\textbf{y})\,h_{2}(\theta)\,\overline{F}_{2}(x)\,,$

that means

$V(\textbf{y})\,h_{2}(\theta)=\varepsilon>0\,,$ (4.31)

hence the dependence (4.24) represents a sub-case of asymptotic independence
for $\Theta$ and X. However, if the vector X has asymptotically independent
components then $\varepsilon=0$. So from (4.31) we conclude either
$V(\textbf{y})=0$ or $h_{2}(\theta)=0$, which is not true by definition.
Namely, the asymptotic independence for $\Theta$ and X fails.

######  Corollary 4.3.

Let ${\bf X}\in MRV(\alpha,\,b,\,\nu)$ be a non-negative random vector and
$\Theta$ be a non-negative random variable. Under the assumptions of Theorem
proposition 4.4 and if the integral in (4.11) converges for some $\zeta>0$,
then for $\Theta\,{\bf X}$ we obtain (4.12).

###### Proof.

As the conditions of proposition 4.4 hold, it follows $\Theta\,{\bf X}\in
MRV(\alpha,\,b,\,\nu)$, and consequently
${\bf\Theta\,X(w)}\in\mathcal{R}_{-\alpha}$, for some $\alpha>0$. Further the
proof follows the route of Theorem 4.1. ∎

##  5\. Conclusion

The asymptotic analysis of the tail distortion risk measures in background
risk model can bring better understanding of the role of the dependence
structures. In fact we intent to approach the real situations, where several
kind of dependencies interact simultaneously.

In fact, the first models with interdependence among $\Theta_{i}$ and $X_{i}$,
namely the main claims $X_{1},\,\ldots,\,X_{n}$ to be dependent, the discount
factors $\Theta_{1}\,,\ldots,\,\Theta_{n}$ to be also dependent and the two
sequences $\\{X_{1},\,\ldots,\,X_{n}\\}$ and
$\\{\Theta_{1}\,,\ldots,\,\Theta_{n}\\}$ to be mutually dependent, are
restricted to the case of multivariate regular variation of main claims. The
GTAI dependence structure in two dimensional models provides a form of
interdependence among $X_{1},\,\ldots,\,X_{n}$ and
$\Theta_{1}\,,\ldots,\,\Theta_{n}$.

Finally, in actuarial practice the randomly stopped sums gain more popularity,
which is plausible with respect to applications.

## References

  * [1] Asimit, A.V., Badescu, A.L. (2010) Extremes on the discounted aggregate claims in a time dependent risk model. Scandinavian Actuarial Journal 2, 93–104.
  * [2] Asimit, A.V., Jones, B.L. (2008) Dependence and the asymptotic behavior of large claims reinsurance. Insurance Math. Econom. 43, no. 3, 407–411.
  * [3] Asimit, V.A., Vernic, R., Zitikis, R. (2016) Background risk models and stepwise portfolio contribution. Methodology and Computing in Applied Probability 18, 805–827.
  * [4] Basrak, B., Davis, R.A., Mikosch, T. (2002) Regular variation of GARCH processes. Stochastic Process. Appl. 99, no. 1, 95–115.
  * [5] Bingham, N.H., Goldie, C.M., Teugels, J.L. (1987) Regular Variation Cambridge University Press, Cambridge.
  * [6] Borovkov, A.A., Borovkov, K.A. (2008) Asymptotic Analysis of Random Walks: Heavy-Tailed distributions Cambridge University Press, Cambridge.
  * [7] Breiman L. (1965) On some limit theorems similar to arc-sin law. Theory Probab. Appl. , 10, 323–331.
  * [8] Cadena, M., Omey, E., Vesilo, R. (2022) Revisiting the product of random variables. J.of Math. Sciences 267, no. 2, 180–195.
  * [9] Chen, Y., Gao, Y., Gao, W.X., Zhang, W.P. (2018) Second order asymptotics of risk concentration of a portfolio with deflated risks. Math. Probability Eng., 4689497, 1–12.
  * [10] Chen, Y., Liu J. (2024) Asymptotic capital allocation based on highr momment risk measure. Eur. Actuar. J., https://doi.org/10.1007/s13385-024-00378-4.
  * [11] Chen, Y., Wang, J., Zhang, W. (2022) Tail distortion risk measure for portfolio with multivariate regular variation. Commun. Math. Stat., 10, 263–285.
  * [12] Chen, Y., Yang, Y. (2019) Bivariate regular variation among randomly weighted sums in general insurance. European Actuarial Journal, 9, 301–322
  * [13] Chen, Y., Yuen, K.C. (2009) Sums of pairwise quasi-asymptotic independent random variables with consistent variation. Stochastic Models, 25, 76–89.
  * [14] Cheng, D., Yu, C. (2019) Uniform asymptotics for the ruin probabilities in a bidimensional risk model with strongly subexponential claims. Stochastics, 91, no.5, 643–656.
  * [15] Cheng, D., Yu, C. (2020) Asymptotic ruin probabilities of a two dimensional renewal risk model with dependent inter-arrival times. Communication in Statistics and Mathematics, 49, no.7, 1742–1760.
  * [16] Cheng, M., Konstantinides, D.G., Wang, D (2022) Uniform asymptotic estimates in a time-dependent risk model with general investment returns and multivariate regularly varying claims. Appl. Math. and Comput., 434, 1–28.
  * [17] Cheng, M., Konstantinides, D.G., Wang, D (2024) Multivariate regularly varying insurance and financial risks in multi-dimensional risk models. J. Appl. Prob., to appear
  * [18] Chistyakov, V.P. (1964) A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Probab. Appl., 9, 640–648.
  * [19] Cline, D.B.H., Samorodnitsky, G. (1994) Subexponentiality of the product of independent random variables. Stoch. Process. Appl., 49, 75–98.
  * [20] Côté, M.P, Genest, C. (2019) Dependence in a background risk model. J Multivatiate Anal., 172, 28–46.
  * [21] de Haan, L., Ferreira, A. (2006) Extreme Value Theory: An Introduction. Springer Verlag, New York.
  * [22] Dhaene, J., Vanduffel, S., Goovaerts, M., Kaas, R., Tang, Q.H. (2006) Risk measures and comonotonicity. Stochastic Models 22, no. 4, 573–606.
  * [23] Drees, H. (1998) On smooth statistical tail functionals. Scand. J. Statist. 25, 187–210.
  * [24] Fougeres, A., Mercadier, C. (2012) Risk measures and multivariate extensions of Breiman’s theorem. J. Appl. Probab., 49, no. 2, 364–384.
  * [25] Geluk, J., Tang, Q. (2009) Asymptotic tail probabilities of sums of dependent subexponential random variables. J. Theoret. Prob., 22, 871–882.
  * [26] Hult, H., Samorodnitsky, G. (2008) Tail probabilities for infinite series of regularly varying vectors. Bernoulli, 14, no. 3, 838–864.
  * [27] Konstantinides, D.G., Li, J. (2016) Asymptotic ruin probabilities for a multidimensional renewal risk model with multivariate regularly varying claims. Insurance Mathematics and Economics, 69, 38–44.
  * [28] Konstantinides, D.G., Passalidis, C.D. (2024) Closure properties and heavy tails: random vectors in the presence of dependence Preprint, arXiv:2402.09041.
  * [29] Leipus, R., Šiaulys, J. (2024) A note on randomly stopped sums with zero mean increments. Modern Stochastics: Theory and Applications, 11, no.1, 31–42.
  * [30] Leipus, R., Šiaulys, J., Konstantinides, D.G. (2023) Closure Properties for Heavy-Tailed and Related Distributions: An Overview. Springer Nature, Cham.
  * [31] Li, J. (2013) On pairwise quasi-asymptotically independent random variables and their applications. Stat. Prob. Lett., 83, 2081–2087.
  * [32] Li, J. (2016) Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return. Insurance Math. Econom., 71, 195–204.
  * [33] Li, J. (2018) On the joint tail behavior of randomly weighted sums of heavy-tailed random variables. J. Multivariate Anal., 164, 40–53.
  * [34] Li, J. (2018(b)) A revisit to asymptotic ruin probabilities of a two-dimensional renewal risk model. Stat. Probab. Lett., 140, 23–32.
  * [35] Li, J. (2022) Asymptotic Analysis of a Dynamic Systemic Risk Measure in a Renewal Risk Model. Insur. Math. Econom., 107, 38–56.
  * [36] Li, J., Tang, Q. (2015) Interplay of insurance and financial risks in a discrete time model with strongly regular variation. Bernoulli 21, no.3, 1800–1823
  * [37] Liu, L. (2009) Precise large deviations for dependent random variables with heavy tails. Statistics and Probability Letters, 79, 1290–1298.
  * [38] Liu, J., Yang, Y. (2021) Asymptotics for systemic risk with dependent heavy-tailed losses. ASTIN Bulletin, 51, no.2, 571–605.
  * [39] Matuszewska, W. (1964) On generalization of regularly increasing functions. Studia Mathematica, 24, 271–279.
  * [40] Maulik, K., Resnick, S.I., Rootzen, H. (2002) Asymptotic independence and a network traffic model. J. Appl. Probab., 39, no.4, 671–699.
  * [41] Olvera-Cravioto, M. (2012) Asymptotic independence and a network traffic model. Adv. Appl. Probab., 44, 1142–1172.
  * [42] Resnick, S. (2007) Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer, New York.
  * [43] Shen, X., Du, K. (2023) Uniform Approximation for the tail behavior of bidimensional randomly weighted sums. Method. Comp. Appl. Prob., no.25
  * [44] Spindys, J., Šiaulys, J. (2020) Regularly distributed randomly stopped sum, minimum and maximum. Non. Anal. Mod. Contr., 25, no.3, 509-522
  * [45] Tang, Q., Tsitsiashvili, G. (2003) Randomly weighted sums with application to ruin theory. Extremes, 6, 171–188
  * [46] Tang, Q., Yuan, Z. (2014) Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes, 17, 467–493
  * [47] Tsanakas A. (2004) Dynamic capital allocation with distortion risk measures. Insurance Mathematics and Economics, 35, no.2, 223–243.
  * [48] Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, 71–92.
  * [49] Yang, Y., Konstantinides D.G. (2015) Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks. Scandinavian Actuarial Journal (8), 641–659.
  * [50] Yang, Y., Leipus, R., Šiaulys, J. (2012(a)) Tail probability of randomly weighted sums of subexponential random variables under a dependence structure. Statistics and Probability Letters, 82, 1727–1736.
  * [51] Yang, Y., Leipus, R., Šiaulys, J. (2012(b)) On the ruin probability in a dependent discrete time risk model with insurance and financial risks. J. Comput. Appl. Mat., 236, 3286–3295.
  * [52] Yang, Y., Leipus, R., Šiaulys, J. (2016) Asymptotics for randomly weighted and stopped dependent sums. Stochastics, 88, no.2, 1–20.
  * [53] Yang, Y., Wang, K., Leipus, R., Šiaulys, J. (2013) A note on the max-sum equivalence of randomly weighted sums of heavy-tailed random variables. Non. Anal. Mod. Contr., 18, no.4, 519–525.
  * [54] Zhou C. (2010) Are banks too big to fail? Measuring systemic importance of financial institutions. International Journal of central banking, 6, 205–250.
  * [55] Zhu, L., Li, H. (2012) Tail distortion risk and its asymptotic analysis. Insurance Math. Econom., 51, no. 1, 115–132.