2. Two-dimensional heavy tails

The reason why the multivariate distributions have been so popular is their ability to describe better multidimensional phenomena. This happens because of the interdependence among the components of the random vectors, which affect significantly the final outcome.

The first heavy-tailed distributions class that was extended to a multidimensional frame is the regular variation. We say that the random vector $\textbf { X}=(X_1,\,\ldots, \,X_d)$ represents a multivariate regularly varying vector with index $\alpha$ and non-degenerate, Radon measure $\nu$ , symbolically $\textbf { X} \in MRV(\alpha, \,F,\,\nu )$ if it holds

\begin{eqnarray*} \lim _{x\to \infty } \dfrac 1{{\overline {F}}(x)}{\mathbf{P}}\left [ \dfrac {\textbf { X}}x \in {\mathbb{B}} \right ]=\nu ({\mathbb{B}}), \end{eqnarray*}

for any $\nu$ -continuous Borel set ${\mathbb{B}} \subset [0,\,\infty ]^d\setminus \{ \textbf { 0}\}$ , with $F \in \mathcal{R}_{-\alpha }$ . The measure $\nu$ is homogeneous; namely, it holds $\nu (\lambda \,{\mathbb{B}}) = \lambda ^{-\alpha }\,\nu ({\mathbb{B}})$ , for any $\lambda \gt 0$ .

The frame of multivariate regular variation was introduced in De Haan and Resnick (Reference De Haan and Resnick1981). Under this definition, the multivariate regular variation was used in the study of several issues in multivariate risk models and in risk management, as, for example, in Li (Reference Li2016), Tang and Yang (Reference Tang and Yang2019), and Yang and Su (Reference Yang and Su2023).

Although this kind of extension to multidimensional setup is well-established, it does not happen to other multidimensional distribution classes. Most of the extensions cover the multivariate subexponential distribution class and the multivariate long-tailed distribution class.

Initially, these two distribution classes were introduced in Cline and Resnick (Reference Cline and Resnick1992) as essential extension of the multivariate regular variation, namely, using vague convergence and point processes. Later, in Omey (Reference Omey2006), three different formulations appear for the multivariate subexponentiality and the multivariate long-tailedness. The formulations, which are close to our definitions, are given in classes $\mathcal{S}({\mathbb{R}}^d)$ and $\mathcal{L}({\mathbb{R}}^d)$ . We say that the multivariate distribution $F$ belongs to class $\mathcal{S}({\mathbb{R}}^d)$ , if it holds

\begin{eqnarray*} \lim _{x\to \infty } \dfrac {\overline {F^{2*}}(\textbf { t}\,x)}{{\overline {F}}(\textbf { t}\,x)}=2, \end{eqnarray*}

for any $\textbf { t} \gt \textbf { 0}$ , with $\min _{1\leq i \leq d} \{t_i\} \lt \infty$ , and that the multivariate distribution $F$ belongs to class $\mathcal{L}({\mathbb{R}}^d)$ , if it holds

\begin{eqnarray*} \lim _{x\to \infty } \dfrac {{\overline {F}}(\textbf { t}\,x-\textbf { a})}{{\overline {F}}(\textbf { t}\,x)}=1, \end{eqnarray*}

for any $\textbf { a} \geq \textbf { 0}$ and for any $\textbf { t} \gt \textbf { 0}$ , with $\min _{1\leq i \leq d} \{t_i\} \lt \infty$ .

This approach was used to study the asymptotic behavior of the tail of a randomly stopped sum of random vectors, namely, $S_{N}=\sum _{i=1}^N \textbf { X}_i$ , where $N$ is a discrete random variable with support ${\mathbb{N}}_0={\mathbb{N}} \cup \{0\}$ and the $ \textbf { X}_i$ are independent, identically distributed random vectors with multivariate distribution $F$ . For applications of this class, see Omey et al. (Reference Omey, Mallor and Santos2006).

Finally, another formulation of multivariate subexponential distributions was provided in Samorodnitsky and Sun (Reference Samorodnitsky and Sun2016), which represents the only approach with results for the ruin probability in a multivariate continuous-time risk model. Although the approach by Samorodnitsky and Sun (Reference Samorodnitsky and Sun2016) is clearly stronger than the previous two, it describes in some sense the linear multivariate single big jump, but it cannot cover the distributions through their joint tail; see Konstantinides and Passalidis (Reference Konstantinides and Passalidis2024, Sec. 5) for comments about this approach, indicating the complementary function to Samorodnitsky and Sun (Reference Samorodnitsky and Sun2016) of our approach, found below. In the present paper, we confine ourselves to the two dimensions, and we stay close to the formulation in Omey (Reference Omey2006); however, we keep two important differences.

First, we follow a direct approach to the one-dimensional distribution classes’ definitions.

Second, in the case of $d=2$ , the formulation in Omey (Reference Omey2006), and in the definition of multivariate regular variation, the convention $\textbf { F}(x,\,y)={\mathbf{P}}[X\leq x,\,Y\leq y]$ is adopted, and the distribution tail $1-\textbf { F}(x,\,y)$ , denoted $\overline {\textbf { F}}(x,\,y)$ , is applied on the event $\{X\gt x\} \cup \{Y\gt y\}$ . We consider only the case in which there exist excesses of both random variables $\{X\gt x\} \cap \{Y\gt y\}$ ; namely, we define by ${ {\overline {\mathbf{F}}}_1}(x,\,y)\,:\!=\,{\mathbf{P}}[X\gt x,\,Y\gt y]$ , as the distribution tail of $\textbf { F}$ , with notation $ { {\overline {\mathbf{F}}}_b}(x,\,y)\,:\!=\,{\mathbf{P}}[X\gt b_1\,x,\,Y\gt b_2\,y]$ , for all $\textbf { b}=(b_1,\,b_2) \in (0,\,\infty )^2$ . The choice of such a definition is due to both the consistency with the univariate case and the ease in asymptotic calculation of the joint tail of random sums as well. We intend that our approach becomes more consistent with the ruin of all portfolios, which represents the worst event that can happen for an insurance company with multiple businesses. In some sense, this is the reason why our classes lead to a nonlinear approach of the single big jump in multidimensional setup.

Next, we introduce the first bivariate heavy-tailed distribution class. From now on and further by the notation $\textbf { a}=(a_1,\,a_2)\gt (0,\,0)$ , we mean that $(a_1,\,a_2) \in [0,\,\infty )^2 \setminus \{\textbf { 0}\}$ , except it is referred to differently.

Remark 2.1. From the previous definition we wonder if by the two-dimensional property of class $\mathcal{L}^{(2)}$ follows directly the inclusion $F,\,G \in \mathcal{L}$ . The answer to this question is no because it holds for any, or equivalently for some, $(a_1,\,a_2) \gt (0,\,0)$ , as follows from Definition 2.1 .

Let $F \in \mathcal{L}$ be a distribution and $G$ be another distribution, not necessarily from class $\mathcal{L}$ . We assume that the two distributions stem from the independent random variables $X$ and $Y$ ; thus, if $a_1\gt 0$ and $a_2=0$ , we find that

\begin{eqnarray*} \lim _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}[X\gt x-a_1, \;Y\gt y]}{{\mathbf{P}}[X\gt x, \;Y\gt y]}=\lim _{x\wedge y \to \infty } \dfrac {{\overline {F}}(x-a_1)\,\overline {G}(y)}{{\overline {F}}(x)\,\overline {G}(y)}=1, \end{eqnarray*}

however, if it holds $G \notin \mathcal{L}$ , then we have not this pair in the class $\mathcal{L}^{(2)}$ .

The reason why we require that the marginals belong to class $\mathcal{L}$ is to secure some two-dimensional closure properties that could fail if the $\mathcal{L}$ condition is missing.

From Definition2.1 we obtain that if $(F,\,G) \in \mathcal{L}^{(2)}$ , then for any $(A_1,\,A_2)\gt (0,\,0)$ , it holds

(2.1) \begin{eqnarray} \sup _{|a_1|\lt A_1,\,|a_2|\lt A_2}\left | {\mathbf{P}}[X\gt x-a_1,\,Y \gt y-a_2]-{\mathbf{P}}[X\gt x,\,Y\gt y]\right |=o\left ( {\mathbf{P}}[X\gt x,\,Y\gt y]\right ), \end{eqnarray}

as $x\wedge y \to \infty$ , which follows from the uniformity of the convergence

\begin{eqnarray*} \lim _{x\wedge y \to \infty }\dfrac { {\mathbf{P}}[X\gt x-a_1,\;Y \gt y-a_2]}{{\mathbf{P}}[X\gt x,\;Y\gt y]}=1, \end{eqnarray*}

over the parallelogram $[\!-A_1,\;A_1]\times [\!-A_2,\;A_2]$ . Indeed, for $-A_1 \leq a_1 \leq A_1$ and $-A_2 \leq a_2 \leq A_2$ , we obtain $x-A_1 \leq x+a_1 \leq x+A_1$ and $y-A_2 \leq y+a_2 \leq y+A_2$ . Hence,

\begin{eqnarray*} \dfrac {{\mathbf{P}}[X\gt x-A_1, \;Y\gt y-A_2]}{{\mathbf{P}}[X\gt x, \;Y\gt y]}&\geq & \dfrac {{\mathbf{P}}[X\gt x+a_1, \;Y\gt y+a_2]}{{\mathbf{P}}[X\gt x, \;Y\gt y]}\\[2mm] &\geq & \dfrac {{\mathbf{P}}[X\gt x+A_1, \;Y\gt y+A_2]}{{\mathbf{P}}[X\gt x, \;Y\gt y]}, \end{eqnarray*}

where the first fraction tends to unity, as $x\wedge y \to \infty$ , by Definition2.1, and the last fraction also tends to unity, as $x\wedge y \to \infty$ , after the change of variables $x'=x+A_1$ and $y'=y+A_2$ and by Definition2.1.

Definition2.2 provides the insensitivity property in joint distributions; see the univariate analogue, for example, in Foss et al. (Reference Foss, Korshunov and Zachary2013) or in Konstantinides (Reference Konstantinides2018).

Definition 2.2. Let $a_F(x),\,a_G(y)\gt 0$ for any $x\gt 0,\,y\gt 0$ be two non-decreasing function. We say that the joint distribution $\textbf { F}=(F,\,G)$ of $(X,\,Y)$ , with right endpoint $r_{\textbf { F}}\,:\!=\,(r_F,\,r_G)=(\infty, \,\infty )$ , satisfies $(a_F,\,a_G)$ -joint insensitivity, if

\begin{eqnarray*} &&\sup _{|a_1|\leq a_F(x),\,|a_2| \leq a_G(y)}\left | {\mathbf{P}}[X\gt x-a_1, Y \gt y-a_2]-{\mathbf{P}}[X\gt x, Y\gt y]\right |\\[2mm] &&\qquad \qquad \qquad =o\left ( {\mathbf{P}}[X\gt x,Y\gt y]\right ), \end{eqnarray*}

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

Now we show that class $ \mathcal{L}^{(2)}$ satisfies the $(a_F,\,a_G)$ -joint insensitive property.

Proof. For any integer $n \in {\mathbb{N}}$ , from relation (2.1), we can choose an increasing to infinity sequence $\{u_n\}$ , such that the inequality

\begin{eqnarray*} \sup _{|a_1|\leq n,\,|a_2| \leq n}\left | {\mathbf{P}}[X\gt x-a_1,\,Y \gt y-a_2]-{\mathbf{P}}[X\gt x,\,Y\gt y]\right |\leq \dfrac {{\mathbf{P}}[X\gt x,\,Y\gt y]}n, \end{eqnarray*}

holds for any $x\geq u_n$ and any $y\geq u_n$ . Without loss of generality, we consider that the sequence $\{u_n\}$ increases to infinity. We put $a_F(x)=a_G(y)=n$ , for any $(x,\,y)\in (u_n,\,u_{n+1}]^2$ . From the fact that $u_n \to \infty$ , as $n\to \infty$ , we obtain that $a_F(x) \to \infty$ , as $x\to \infty$ , and $a_G(y) \to \infty$ , as $y\to \infty$ .

So, from the construction of $a(\cdot )$ , we conclude that

\begin{eqnarray*} \sup _{|a_1|\leq a_F(x),\,|a_2| \leq a_G(y)}\left | {\mathbf{P}}[X\gt x-a_1,\,Y \gt y-a_2]-{\mathbf{P}}[X\gt x,\,Y\gt y]\right |\leq \dfrac {{\mathbf{P}}[X\gt x,\,Y\gt y]}{n}, \end{eqnarray*}

for any $x \gt u_n$ and any $y\gt u_n$ , which is the required result.

Remark 2.2. From the $(a_F,\,a_G)$ -joint insensitivity, it does not follow necessarily that $a_F$ and $a_G$ are insensitivity functions for the marginal distributions $F,\,G$ , respectively. Furthermore, Lemma 2.1 asserts that

\begin{eqnarray*} \lim _{x\wedge y \to \infty }\dfrac { {\mathbf{P}}[X\gt x\pm a_F(x),\;Y \gt y \pm a_G(y)]}{{\mathbf{P}}[X\gt x,\;Y\gt y]}=1. \end{eqnarray*}

Let us see now two examples that help either to understanding or to constructing of such bivariate distributions. The first case is the simplest, as we construct $(X,\,Y) \in \mathcal{L}^{(2)}$ through the independence between $X$ and $Y$ .

Example 2.1. Let $X$ and $Y$ be random variables with distributions $F \in \mathcal{L}$ and $G \in \mathcal{L}$ , respectively. We assume that $X$ and $Y$ are independent, to obtain

\begin{eqnarray*} \lim _{x\wedge y \to \infty } \dfrac {{ {\overline {\mathbf{F}}}_1}(x-a_1,\,y-a_2)}{ { {\overline {\mathbf{F}}}_1}(x,\,y)}&=&\lim _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}[X\gt x-a_1,\,Y\gt y-a_2]}{{\mathbf{P}}[X\gt x,\,Y\gt y]}\\[2mm] &=&\lim _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}[X\gt x-a_1]}{{\mathbf{P}}[X\gt x]}\,\dfrac {{\mathbf{P}}[Y\gt y-a_2]}{{\mathbf{P}}[Y\gt y]}=1. \end{eqnarray*}

Therefore $(X,\,Y) \in \mathcal{L}^{(2)}$ .

The next example makes sense, as it cannot be reduced into univariate distributions. The following dependence structure can be found in Li (Reference Li2018). We say that the random variables $X$ and $Y$ are strongly asymptotic independent (SAI) if ${\mathbf{P}}[X^- \gt x,\,Y\gt y]=O[F(\!-x)\,\overline {G}(y)]$ , ${\mathbf{P}}[X \gt x,\,Y^-\gt y]=O[{\overline {F}}(x)\,G(\!-y)]$ hold as $x\wedge y \to \infty$ , and there exists a constant $C\gt 0$ such that if it holds

(2.2) \begin{eqnarray} {\mathbf{P}}[X\gt x,\,Y\gt y] \sim C\,{\overline {F}}(x)\,\overline {G}(y), \end{eqnarray}

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

If the $X$ and $Y$ are bounded from below, then (2.2) is enough to be SAI.

Example 2.2. Let $X$ and $Y$ be random variables with strongly asymptotic independence, with some constant $C\gt 0$ and distributions $F \in \mathcal{L}$ and $G \in \mathcal{L}$ , respectively. Then

\begin{eqnarray*} \lim _{x\wedge y \to \infty } \dfrac {{ {\overline {\mathbf{F}}}_1}(x-a_1,\,y-a_2)}{ { {\overline {\mathbf{F}}}_1}(x,\,y)}&=&\lim _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}[X\gt x-a_1,\,Y\gt y-a_2]}{{\mathbf{P}}[X\gt x,\,Y\gt y]}\\[2mm] &=&\lim _{x\wedge y \to \infty }\dfrac {C\,{\overline {F}}(x-a_1)\,\overline {G}(y-a_2)}{C\,{\overline {F}}(x)\,\overline {G}(y)}=1. \end{eqnarray*}

Therefore $(X,\,Y) \in \mathcal{L}^{(2)}$ .

The first two examples restrict themselves either in the independent case or in some kind of asymptotic independence. Notice that in the next example as class $ \mathcal{L}^{(2)}$ , we understand the class from Definition2.1, but with a restriction with respect to convergence, instead of $x \wedge y$ to $x=y$ only. In Li and Yang (Reference Li and Yang2015), the dependence structure from relation (2.3) was used, through the survival copula $\widehat {C}$ , to depict the dependence relation among claims in a bivariate, continuous-time risk model. We assume that for two random variables $X,\,Y$ following a survival copula $\widehat {C}$ , there exists some constant $\gamma \geq 1$ and a positive measurable function $h(\cdot, \,\cdot )$ , such that the asymptotic relation holds

(2.3) \begin{eqnarray} \widehat {C}(t_1\,x,\,t_2\,x) \sim x^{\gamma }\,h(t_1,\,t_2), \end{eqnarray}

as $x \downarrow 0$ holds, for any $(t_1,\,t_2) \in (0,\,\infty )$ .

Example 2.3. Let the random variables $X,\,Y$ follow a survival copula from relation (2.3) and $F,\,G$ be their marginal distributions. Furthermore, we assume that it holds

(2.4) \begin{eqnarray} \lim _{x\to \infty } \dfrac {\overline {G}(x)}{{\overline {F}}(x)}=c, \end{eqnarray}

for some positive constant $c\gt 0$ and either $F\in \mathcal{L}$ or $G\in \mathcal{L}$ is true. Finally, we suppose that relation (2.3) holds with $\gamma =1$ . Then we obtain $F,\,G \in \mathcal{L}$ , which follows from the closure property of class $ \mathcal{L}$ with respect to strong equivalence of (2.4); see Leipus et al. (Reference Leipus, Šiaulys and Konstantinides2023). From Li and Yang (Reference Li and Yang2015, Prop. 3.1), we have the random variables $X,\,Y$ to be asymptotic dependent, and further, they satisfy

\begin{eqnarray*} \lim _{x\to \infty } \dfrac {{\mathbf{P}}[X\gt x,\,Y\gt x]}{{\mathbf{P}}[X\gt x]} =h(1,\,c)\gt 0, \end{eqnarray*}

hence by the last formulas, for any $(a_1,\,a_2)\gt (0,\,0)$ , it holds

\begin{eqnarray*} \lim _{x\to \infty } \dfrac {{\mathbf{P}}[X\gt x-a_1,\,Y\gt x-a_2]}{{\mathbf{P}}[X\gt x,\,Y\gt x]} =\lim _{x\to \infty } \dfrac {h(1,\,c)\,{\mathbf{P}}[X\gt x-a_1]}{h(1,\,c)\,{\mathbf{P}}[X\gt x]}=1, \end{eqnarray*}

so we find $ (X,\,Y) \in \mathcal{L}^{(2)}$ , in the sense that in Definition 2.1 , the convergence is valid with $x=y$ .

We can find several dependence structures that satisfy the $\mathcal{L}^{(2)}$ condition. However, we choose to pursue theoretical results.

Now we pass to the bivariate subexponential distribution class $\mathcal{S}^{(2)}$ .

Remark 2.3. In case of $d$ -variate distribution, relation (2.5) becomes

\begin{eqnarray*} \lim _{x_1\wedge \ldots \wedge x_n\; to\; \infty} \dfrac {{\mathbf{P}}[X_{1,1}+X_{1,2}\gt x_1,\;\ldots, \,X_{d,1}+X_{d,2}\gt x_d]}{{\mathbf{P}}[X_{1,1}\gt x_1,\;\ldots, \,X_{d,1}\gt x_d]}=2^d. \end{eqnarray*}

Now we come to the bivariate dominatedly varying distribution class $\mathcal{D}^{(2)}$ .

It is obvious that 2.6 is equivalently with:

\begin{eqnarray*} \liminf _{x\wedge y \to \infty }\dfrac { { {\overline {\mathbf{F}}}_b}(x,\, y)}{ { {\overline {\mathbf{F}}}_1}(x, y)}\gt 0 \end{eqnarray*}

for some, or equivalently for all $\textbf { b}=(b_1,\,b_2) \in (1,\,\infty )^2$ , with $b_1$ not necessarily equal to $b_2$ .

Remark 2.4. In Konstantinides and Passalidis (Reference Konstantinides and Passalidis2024b), the class $\mathcal{D}_{n}$ (for some $n\in \mathbb{N}$ ) of multivariate dominatedly varying random vectors was introduced. It is obvious that in case $n=2$ , our approach includesthis definition. Specifically

\begin{equation*} \mathcal{D}_{2}\subset \mathcal{D}^{(2)} \end{equation*}

Examples2.1 and 2.2 remain intact in classes $\mathcal{D}^{(2)}$ and $\mathcal{C}^{(2)}$ ; hence, they keep functioning in class $(\mathcal{D}\cap \mathcal{L})^{(2)}\,:\!=\,\mathcal{D}^{(2)} \cap \mathcal{L}^{(2)}$ .

Proof. Let consider that $(F,\,G) \in \mathcal{C}^{(2)}$ . Then, for $\textbf { a}=(a_1,\,a_2)>(0,\,0)$ for any distributions $F,\,G$ , we obtain

(2.7) \begin{eqnarray} 1\leq \liminf _{x \wedge y \to \infty } \dfrac {{\mathbf{P}}[X\gt x-a_1,\,Y\gt y-a_2]}{{\mathbf{P}}[X\gt x,\,Y\gt y]}. \end{eqnarray}

Hence, we have to show that the upper bound of the last fraction is equal to unity. We observe that for any small enough $\delta _1,\,\delta _2 \gt 0$ , there exist some $x_0\gt 0$ , such that $x\,(1-\delta _1) \leq x- a_1$ and $y\,(1-\delta _2) \leq y- a_2$ , for any $x \wedge y \geq x_0$ . Therefore, we find

(2.8) \begin{eqnarray} \limsup _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}[X\gt x-a_1, Y\gt y-a_2]}{{\mathbf{P}}[X\gt x, Y\gt y]}\leq \limsup _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}[X\gt x (1-\delta _1),\,Y\gt y (1-\delta _2)]}{{\mathbf{P}}[X\gt x,\,Y\gt y]}\to 1, \end{eqnarray}

as $(\delta _1,\,\delta _2) \to (0,\,0)$ , where in the last step, we use the properties of class $\mathcal{C}^{(2)}$ for the pair of distributions $(F,\,G)$ . So, by relations (2.7) and (2.8), we conclude that $(X,\,Y) \in \mathcal{L}^{(2)}$ .

3. Max-sum equivalence and closure properties with respect to convolution

Now, we present two definitions. In the first one, we define the closure property with respect to convolution in bivariate distributions. In this case, we formulate the main result, showing that the class $\mathcal{D}^{(2)}$ is closed. The second definition, given at the end of the section, under concrete dependence structures, also presented later, is fulfilled with respect to classes $(\mathcal{D}\cap \mathcal{L})^{(2)}$ and $\mathcal{C}^{(2)}$ .

In the last definition, although the check of $F_{X_1+X_2} \in \mathcal{B}$ , $G_{Y_1+Y_2} \in \mathcal{B}$ is implied directly by the univariate closure properties, the check of $(F_k,\,G_l) \in \mathcal{B}^{(2)}$ , for any $k,\,l \in \{1,\,2\}$ , is still NOT implied. Also, it is NOT implied that the joint tail of $(X_1+X_2,\,Y_1+Y_2)$ has the desired property of $\mathcal{B}^{(2)}$ . Hence, we find out that the dependence structures among the components play a crucial role in the closure properties of bivariate vectors.

Next we see that the class $\mathcal{D}^{(2)}$ is closed with respect to sum (of arbitrarily dependent random vectors with arbitrarily non-negative dependent components), under the condition that the point $(1)$ in Definition3.1 is satisfied.

Proof. At first, for the first condition of $\mathcal{D}^{(2)}$ , we obtain $F_{X_1+X_2} \in \mathcal{D}$ and $G_{Y_1+Y_2} \in \mathcal{D}$ because of Proposition1.1.

Taking into consideration that all the distributions have support on the interval $[0,\,\infty )$ , by the elementary inequalities

\begin{eqnarray*} {\mathbf{P}}[X_1+X_2\gt x] \leq {\mathbf{P}}\left [X_1\gt \dfrac x2 \right ]+ {\mathbf{P}}\left [X_2 \gt \dfrac x2\right ],\\[2mm] {\mathbf{P}}[X_1+X_2\gt x] \geq \dfrac 12 \left ({\mathbf{P}}[X_1\gt x]+ {\mathbf{P}}[X_2 \gt x]\right ), \end{eqnarray*}

we find

\begin{eqnarray*} {\mathbf{P}}[X_1+X_2\gt x,\,Y_1+Y_2\gt y] &\leq & {\mathbf{P}}[X_1\gt x/2,\,Y_1+Y_2\gt y]+ {\mathbf{P}}[X_2\gt x/2,\,Y_1+Y_2\gt y]\\[2mm] &\leq & {\mathbf{P}}\left [X_1\gt \dfrac x2,\,Y_1\gt \dfrac y2\right ]+ {\mathbf{P}}\left [X_1\gt \dfrac x2,\,Y_2\gt \dfrac y2\right ]\\[2mm] &&+{\mathbf{P}}\left [X_2\gt \dfrac x2,\,Y_1\gt \dfrac y2\right ]+{\mathbf{P}}\left [X_2\gt \dfrac x2,\,Y_2\gt \dfrac y2\right ], \end{eqnarray*}

hence

(3.1) \begin{eqnarray} {\mathbf{P}}[X_1+X_2\gt x,\,Y_1+Y_2\gt y]\leq \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt \dfrac x2,\,Y_l\gt \dfrac y2\right ]. \end{eqnarray}

From the other side

\begin{eqnarray*} &&{\mathbf{P}}[X_1+X_2\gt x,\,Y_1+Y_2\gt y] \geq \dfrac {{\mathbf{P}}[X_1\gt x,\,Y_1+Y_2\gt y]+ {\mathbf{P}}[X_2\gt x,\,Y_1+Y_2\gt y]}2 \\[2mm] &&\geq \dfrac {{\mathbf{P}}[X_1\gt x,\,Y_1\gt y]+ {\mathbf{P}}[X_1\gt x,\,Y_2\gt y]+{\mathbf{P}}[X_2\gt x,\,Y_1\gt y]+{\mathbf{P}}[X_2\gt x,\,Y_2\gt y]}4, \end{eqnarray*}

from where we obtain

(3.2) \begin{eqnarray} {\mathbf{P}}[X_1+X_2\gt x,\,Y_1+Y_2\gt y]\geq \dfrac 14\,\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}[X_k\gt x,\,Y_l\gt y]. \end{eqnarray}

Therefore by relations (3.1) and (3.2), due to $(X_k,\,Y_l) \in \mathcal{D}^{(2)}$ for any $k,\,l\,\in \{1,\,2\}$ , and $\textbf { b}=(b_1,\,b_2) \in (0,\,1)^2$ , we find

\begin{eqnarray*} &&\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}[X_1+X_2\gt b_1\,x,\,Y_1+Y_2\gt b_2\,y]}{{\mathbf{P}}[X_1+X_2\gt x,\,Y_1+Y_2\gt y]}\\[2mm] &&\qquad \qquad \leq 4\,\limsup _{x\wedge y \to \infty }\dfrac {\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt \dfrac {b_1}2\,x,\,Y_l\gt \dfrac {b_2}2\,y\right ]}{\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}[X_k\gt x,\,Y_l\gt y]}\\[2mm] &&\qquad \qquad \leq 4\,\max _{k,\,l\,\in \{1,\,2\}}\left \{ \limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_k\gt \dfrac {b_1}2\,x,\,Y_l\gt \dfrac {b_2}2\,y\right ]}{{\mathbf{P}}[X_k\gt x,\,Y_l\gt y]}\right \} \lt \infty . \end{eqnarray*}

So we conclude $(X_1+X_2,\,Y_1+Y_2) \in \mathcal{D}^{(2)}$ .

Now we are ready to define the max-sum equivalence in two dimensions.

Definition 3.2. Let $X_1,\,X_2,\,Y_1,\,Y_2$ be random variables. Then we say that they are joint max-sum equivalent if

\begin{eqnarray*} {\mathbf{P}}[X_1+X_2\gt x,\,Y_1+Y_2\gt y]\sim \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}[X_k\gt x,\,Y_l\gt y], \end{eqnarray*}

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

This kind of asymptotic relation will be established for classes $(\mathcal{D}\cap \mathcal{L})^{(2)}$ and $\mathcal{C}^{(2)}$ , under the assumption of some specific dependence structure.

4. Joint behavior of random sums

In one dimension, the following asymptotic relation attracted attention:

(4.1) \begin{eqnarray} {\mathbf{P}}\left [ \sum _{i=1}^n X_i\gt x \right ]\sim \sum _{i=1}^n {\mathbf{P}}[X_i\gt x], \end{eqnarray}

as $x\to \infty$ . Therefore, we study the behavior of both the maximum $\bigvee _{i=1}^n X_i$ and the maximum of sums

\begin{eqnarray*} \bigvee _{i=1}^n S_i\,:\!=\,\max _{1\leq k \leq n}\sum _{i=1}^k X_i, \end{eqnarray*}

for some distributions and correspondingly with some dependence structures to examine if it holds

(4.2) \begin{eqnarray} {\mathbf{P}}\left [\sum _{i=1}^n X_i\gt x\right ] \sim {\mathbf{P}}\left [ \bigvee _{i=1}^n X_i\gt x \right ]\sim {\mathbf{P}}\left [ \bigvee _{i=1}^n S_i\gt x \right ]\sim \sum _{i=1}^n{\mathbf{P}}\left [ X_i\gt x \right ], \end{eqnarray}

as $x\to \infty$ . Relations (4.1) and (4.2) have been studied extensively; see, for example, in Geluk and Ng (Reference Geluk and Ng2006), Geluk and Tang (Reference Geluk and Tang2009), Ng et al. (Reference Ng, Tang and Yang2002), and Jiang et al. (Reference Jiang, Gao and Wang2014). A similar interest has been appeared for weighted sums of the form

\begin{eqnarray*} S_n^{\Theta } \,:\!=\, \sum _{i=1}^n \Theta _i\,X_i,\qquad \bigvee _{i=1}^n S_i^{\Theta }\,:\!=\,\max _{1\leq k \leq n} \sum _{i=1}^k \Theta _i\,X_i, \end{eqnarray*}

and for the circumstances when they satisfy relations (4.1) and (4.2), see, for example, Tang and Yuan (Reference Tang and Yuan2014), Tang and Tsitsiashvili (Reference Tang and Tsitsiashvili2003), Yang et al. (Reference Yang, Leipus and Šiaulys2012), and Zhang et al. (Reference Zhang, Shen and Weng2009).

In this section, we study relation (4.2) in two dimensions. This can be achieved for the class $(\mathcal{D}\cap \mathcal{L})^{(2)}$ under generalized tail asymptotic independence (GTAI). Although the univariate randomly weighted sums are well studied, this is not true for the multivariate case.

Let us mention some papers involved in the asymptotic behavior of the joint-tail probability

\begin{eqnarray*} {\mathbf{P}}\left [ \sum _{i=1}^n \Theta _i\,X_i \gt x,\;\sum _{j=1}^n \Delta _i\,Y_i \gt y \right ], \end{eqnarray*}

as, for example, Chen and Yang (Reference Chen and Yang2019), Li (Reference Li2018), Shen and Du (Reference Shen and Du2023), Shen et al. (Reference Shen, Ge and Fu2020), and Yang et al. (Reference Yang, Chen and Yuen2024).

We restrict ourselves at the moment in the study of non-weighted random sums of the following form:

\begin{eqnarray*} {\mathbf{P}}\left [ \sum _{i=1}^n X_i\gt x,\;\sum _{j=1}^n Y_j \gt y \right ]. \end{eqnarray*}

Let us remind that, as before, the $X_i$ , $Y_j$ follow distributions with infinite right endpoint.

We note that, in almost all existing papers, the dependence structure for the main variables $X_i$ , $Y_j$ is either of the form: $\{(X_i,\,Y_i),\;i \in {\mathbb{N}}\}$ independent random vectors, and there exists some dependence structure in each random pair, or there exists dependence among $X_1,\,\ldots, \,X_n$ and $Y_1,\,\ldots, \,Y_n$ , but the $X_i$ and $Y_j$ are independent for any $i,\,j$ . Using GTAI, introduced in Konstantinides and Passalidis (Reference Konstantinides and Passalidis2024), both dependence structures are simultaneously permitted. GTAI is defined as follows. Let us consider two sequences of random variables $\{X_n,\;n\in {\mathbb{N}}\},\; \{Y_m,\;m\in {\mathbb{N}}\}$ . We say that the random variables $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_m$ follow the GTAI, if

1. (1) It holds

   \begin{eqnarray*} \lim _{\min \{x_i,\,x_k,\,y_j\} \to \infty } {\mathbf{P}}[|X_i|\gt x_i\;|\;X_k\gt x_k,\, Y_j \gt y_j]=0, \end{eqnarray*}
   for any $1\leq i \neq k \leq n$ , $j=1,\,\ldots, \,m$ .

2. (2) It holds

   \begin{eqnarray*} \lim _{\min \{x_i,\,y_k,\,y_j\} \to \infty } {\mathbf{P}}[|Y_j|\gt y_j\;|\;X_i\gt x_i,\, Y_k \gt y_k]=0, \end{eqnarray*}
   for any $1\leq j \neq k \leq m$ , $i=1,\,\ldots, \,n$ .

The aim of this dependence structure is to model the dependence both within each sequence of random variables and the interdependence between the sequences. We have to notice that if the $X_i$ and $Y_j$ are independent for any $i,\,j$ , then each sequence of random variables follows tail asymptotic dependence (TAI) (see definition below); however, in any other case, the GTAI does not restrict each sequence to TAI, but in a more general form of dependence.

It is easy to find that GTAI contains the case when $X_1,\,\ldots, \,X_n$ are independent or when $Y_1,\,\ldots, \,Y_n$ are independent or both. Even more, this dependence structure indicates that the probability to happen three extreme events is negligible with respect to the probability to happen two extreme events, one in each sequence, and in some sense, $GTAI$ belongs to the dependencies of second-order asymptotic independence.

In most of our results, we use the TAI dependence structure as an extra assumption, which characterizes the dependence of the terms of each sequence. This dependence structure was introduced by Geluk and Tang (Reference Geluk and Tang2009). We say that $X_1,\,\ldots, \,X_n$ are tail asymptotic independent, symbolically $TAI$ (and sometimes named strong quasi-asymptotically independent), if for any pair $i,\,j =1,\,\ldots, \,n$ , with $i \neq j$ , it holds the limit

\begin{eqnarray*} \lim _{x_i \wedge x_j \to \infty }{\mathbf{P}}[ |X_i| \gt x_i\;|\; X_j \gt x_j] = 0. \end{eqnarray*}

The next result provides an asymptotic relation for the maximum of two sequences of random variables under the GTAI, WITHOUT imposing any assumption on the distributions of $ X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_m$ (except the infinite right point).

Theorem 4.1. If $X_1,\,\ldots, \,X_n$ are random variables with distributions $F_1,\,\ldots, \,F_n$ , respectively, and $Y_1,\,\ldots, \,Y_m$ are random variables with distributions $G_1,\,\ldots, \,G_m$ and $X_1,\,\ldots, \,X_n$ , $Y_1,\,\ldots, \,Y_m$ are GTAI, it holds

\begin{eqnarray*} {\mathbf{P}}\left [\bigvee _{i=1}^n X_i\gt x,\;\bigvee _{j=1}^m Y_j\gt y \right ]\sim \sum _{i=1}^n \sum _{j=1}^m {\mathbf{P}}\left [ X_i\gt x,\; Y_j\gt y \right ], \end{eqnarray*}

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

Proof. For $x\gt 0,\,y\gt 0$ holds

(4.3) \begin{eqnarray} {\mathbf{P}}\left [\bigvee _{i=1}^n X_i\gt x,\;\bigvee _{j=1}^m Y_j\gt y \right ]\leq \sum _{i=1}^n \sum _{j=1}^m {\mathbf{P}}\left [ X_i\gt x,\; Y_j\gt y \right ]. \end{eqnarray}

Further for the lower bound, we use Bonferroni’s inequality

\begin{eqnarray*} &&{\mathbf{P}}\left [\bigvee _{i=1}^n X_i\gt x,\;\bigvee _{j=1}^m Y_j\gt y \right ] \\[2mm] &&\geq \sum _{i=1}^n {\mathbf{P}}\left [ X_i\gt x,\; \bigvee _{j=1}^m Y_j\gt y \right ]- {\sum \sum }_{ i\lt l =1} ^{n} {\mathbf{P}}\left [ X_i\gt x,\;X_l\gt x,\: \bigvee _{j=1}^m Y_j\gt y \right ] \\[2mm] &&\geq \sum _{i=1}^n \sum _{j=1}^m {\mathbf{P}}\left [ X_i\gt x,\; Y_j\gt y \right ]- \sum _{i=1}^n {\sum \sum }_{j\lt k =1}^m {\mathbf{P}}\left [ X_i\gt x,\;Y_j\gt y,\; Y_k\gt y \right ]\\[2mm] &&-{\sum \sum }_{l\lt i=1}^n \sum _{j=1}^m {\mathbf{P}}\left [ X_i\gt x,\;X_l\gt x,\; Y_j\gt y \right ]\,=\!:\, I_1(x,\,y) - I_2(x,\,y) - I_3(x,\,y). \end{eqnarray*}

For $I_2(x,\,y)$ , we obtain

\begin{eqnarray*} I_2(x,\,y)&=&\sum _{i=1}^n {\sum \sum }_{j\lt k =1}^m {\mathbf{P}}\left [ X_i\gt x,\;Y_j\gt y,\; Y_k\gt y \right ]\\[2mm] &=&\sum _{i=1}^n {\sum \sum }_{j \lt k=1}^m {\mathbf{P}}\left [ Y_k\gt y\;|\; X_i\gt x,\;Y_j\gt y \right ]\, {\mathbf{P}}\left [ X_i\gt x,\;Y_j\gt y\right ]\\[2mm] &=&o\left ( \sum _{i=1}^n \sum _{j=1}^m {\mathbf{P}}\left [ X_i\gt x,\;Y_j\gt y \right ]\right )=o[I_1(x,\,y)], \end{eqnarray*}

as $x\wedge y \to \infty$ , where in the last step, we use the GTAI property. In a similar way, we can find

\begin{eqnarray*} I_3(x,\,y)=o[I_1(x,\,y)], \end{eqnarray*}

as $x\wedge y \to \infty$ . Hence we conclude

(4.4) \begin{eqnarray} {\mathbf{P}}\left [\bigvee _{i=1}^n X_i\gt x,\;\bigvee _{j=1}^m Y_j\gt y \right ]\gtrsim \sum _{i=1}^n \sum _{j=1}^m {\mathbf{P}}\left [ X_i\gt x,\; Y_j\gt y \right ], \end{eqnarray}

as $x\wedge y \to \infty$ . Now, from relations (4.3) and (4.4), we have the result.

Before next theorem, we need some preliminary lemmas. The next lemma provides an important property of the GTAI structure, presenting itself as closure property with respect to sum.

Lemma 4.1. If $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_m$ follow the GTAI, then holds

(4.5) \begin{eqnarray} \lim _{\min \{x_I,\,x_k,\,y_j\} \to \infty } {\mathbf{P}}\left [\left | \sum _{i \in I} X_i \right |\gt x_I\;\Big |\;X_k\gt x_k,\; Y_j\gt y_j \right ] =0, \end{eqnarray}

for $I\subsetneq \{1,\,\ldots, \,n\}$ and $k \in \{1,\,\ldots, \,n\} \setminus I$ , $j=1,\,\ldots, \,m$ . Similarly holds

(4.6) \begin{eqnarray} \lim _{\min \{x_i,\,y_k,\,y_J\} \to \infty } {\mathbf{P}}\left [\left | \sum _{j \in J} Y_j \right |\gt y_J\;\Big |\;Y_k\gt y_k,\; X_i\gt x_i \right ] =0, \end{eqnarray}

for $J\subsetneq \{1,\,\ldots, \,m\}$ and $k \in \{1,\,\ldots, \,m\} \setminus J$ , $i=1,\,\ldots, \,n$ .

Proof. It is enough to show relation (4.5) as relation (4.6) follows by similar way. Indeed, we observe that

\begin{eqnarray*} &&\lim _{\min \{x_I,\,x_k,\,y_j\} \to \infty } {\mathbf{P}}\left [\left | \sum _{i \in I} X_i \right |\gt x_I\;\Big |\;X_k\gt x_k,\; Y_j\gt y_j \right ] \\[2mm] &&\qquad \qquad \leq \lim _{\min \{x_I,\,x_k,\,y_j\} \to \infty } \sum _{i \in I}{\mathbf{P}}\left [\left | X_i \right |\gt \dfrac {x_I}n\;\Big |\;X_k\gt x_k,\; Y_j\gt y_j \right ]=0, \end{eqnarray*}

where the last step follows from $GTAI$ property.

In most of the following results, we assume that the random variables $X_1,\,\ldots, \,X_n$ , $Y_1,\,\ldots, \,Y_n$ are $GTAI$ and follow distributions from some class $ \mathcal{B} \in \{ \mathcal{C},\, \mathcal{D}\cap \mathcal{L},\, \mathcal{L}\}$ , and at the same time, it holds $(X_k,\,Y_l) \in \mathcal{B}^{(2)}$ , for any $k,\,l \in \{1,\,\ldots, \,n\}$ . Following a referee’s advice, we provide some examples to show that $GTAI$ and $(X_k,\,Y_l) \in \mathcal{B}^{(2)}$ are possible simultaneously. For the sake of simplicity, we consider only the case $n=2$ with non-negative random variables.

Example 4.1. Let $X_1,\,X_2,\,Y_1,\,Y_2$ be non-negative random variables with distributions from class $ \mathcal{B} \in \{ \mathcal{C},\, \mathcal{D}\cap \mathcal{L},\, \mathcal{L}\}$ . Further, we suppose that the $X_1,\,X_2$ are $TAI$ , the $Y_1,\,Y_2$ are also $TAI$ , while the $(X_1,\,X_2)$ and $(Y_1,\,Y_2)$ are independent random pairs. Then, we directly find that $(X_k,\,Y_l) \in \mathcal{B}^{(2)}$ , and additionally by $TAI$ , we obtain that for any $\varepsilon \gt 0$ , there exists some $x_0\gt 0$ , such that for any $1\leq i \neq k \leq 2$ , it holds ${\mathbf{P}}[X_i\gt x_i\;|\;X_k \gt x_k] \lt \varepsilon$ , for any $x_i\wedge x_k \geq x_0$ . Hence for any $1\leq i \neq k \leq 2$ , $j=1,\,2$ , we conclude

\begin{eqnarray*} {\mathbf{P}}\left [X_i\gt x_i,\; X_k\gt x_k, \;Y_j\gt y_j\right ]&=& {\mathbf{P}}\left [X_i\gt x_i,\; X_k\gt x_k\right ]\,{\mathbf{P}}\left [Y_j\gt y_j\right ] \\[2mm] &\lt & \varepsilon \,{\mathbf{P}}\left [X_k\gt x_k\right ]\,{\mathbf{P}}\left [Y_j\gt y_j\right ] = \varepsilon \,{\mathbf{P}}\left [X_k\gt x_k,\,Y_j\gt y_j\right ], \end{eqnarray*}

for any $x_i\wedge x_k \geq x_0$ . From the last relation, because of the arbitrary choice of $\varepsilon$ , we get ${\mathbf{P}}\left [X_i\gt x_i\;|\; X_k\gt x_k, \;Y_j\gt y_j\right ] \to 0$ , as $x_i\wedge x_k \wedge y_j \to \infty$ .

Similarly, by symmetry, we obtain for any $1\leq j \neq k \leq 2$ , $i=1,\,2$ the convergence ${\mathbf{P}}\left [Y_j\gt y_j\;|\; X_i\gt x_i, \;Y_k\gt y_k\right ] \to 0$ , as $x_i\wedge y_k \wedge y_j \to \infty$ . Hence, the $X_1,\,X_2,\,Y_1,\,Y_2$ satisfy the $GTAI$ .

Example 4.2. Let $X_1,\,X_2,\,Y_1,\,Y_2$ be non-negative random variables with distributions from class $ \mathcal{B} \in \{ \mathcal{C},\, \mathcal{D}\cap \mathcal{L},\, \mathcal{L}\}$ . We suppose that $Z_i,\,Z_j,\,Z_k \in \{X_1,\,X_2,\,Y_1,\,Y_2\}$ with $Z_i \neq Z_j \neq Z_k$ and $z_i,\,z_j,\,z_k \in \{x_1,\,x_2,\,y_1,\,y_2\}$ , where the $z_i,\,z_j,\,z_k$ are allowed to be equal. Let us assume the $SAI$ property for any duo or trio of them, namely

\begin{eqnarray*} {\mathbf{P}}[Z_i\gt z_i,\;Z_j\gt z_j] \sim C_{ij}\,{\mathbf{P}}[Z_i\gt z_i]\,{\mathbf{P}}[Z_j\gt z_j], \end{eqnarray*}

as $z_i\wedge z_j \to \infty$ , with $C_{ij} \gt 0$ and

\begin{eqnarray*} {\mathbf{P}}[Z_i\gt z_i,\;Z_j\gt z_j,\;Z_k\gt z_k] \sim C_{ijk}\,{\mathbf{P}}[Z_i\gt z_i]\,{\mathbf{P}}[Z_j\gt z_j]\,{\mathbf{P}}[Z_k\gt z_k], \end{eqnarray*}

as $z_i\wedge z_j\wedge z_k \to \infty$ , with $C_{ijk} \gt 0$ . Then, by $SAI$ in duo mode, we obtain $(X_k,\,Y_l) \in \mathcal{B}^{(2)}$ , for $k,\,l \in \{1,\,2\}$ (see Example 2.2 , for the case of class $\mathcal{L}$ ). Next, by $SAI$ in trio mode, we obtain directly the $GTAI$ .

From here on, we study only the case $n=m$ . In the next lemma, we find the lower asymptotic bound of the joint tail of the random sums

\begin{eqnarray*} S_n \,:\!=\, \sum _{k=1}^n X_k,\qquad \qquad T_n \,:\!=\, \sum _{l=1}^n Y_l, \end{eqnarray*}

when the summands follow distributions with long tails and the $\mathcal{L}^{(2)}$ property is true for any pair of the summands distribution. A similar result, for the unidimensional case, can be found in Geluk and Tang (Reference Geluk and Tang2009), where the dependence structure is TAI. In the next result, we find generalization to two dimensions and furthermore the GTAI assumption. Next, we introduce the notations

\begin{eqnarray*} S_{n,k}\,:\!=\,S_n- X_k,\qquad T_{n,l}\,:\!=\,T_n- Y_l, \end{eqnarray*}

for some $k\in \{1,\,\ldots, \,n\}$ and some $l\in \{1,\,\ldots, \,n\}$ . In what follows, we can choose

(4.7) \begin{eqnarray} a=(a_F,\,a_G)\,:\!=\,\left (\min _{1\leq k \leq n}a_{F_k},\;\min _{1\leq l \leq n} a_{G_l} \right ), \end{eqnarray}

namely, the minimum of all the joint insensitivity functions, which means that the function $a(\cdot )$ is insensitive for all the distribution pairs $(F_k,\,G_l)$ , for $k,\,l \in \{1,\,\ldots, \,n\}$ . In what follows, for the sake of simplicity, the function $a(\cdot )$ is understood either as $a_{F}$ for the $X_k$ or as $a_{G}$ for the $Y_l$ .

Lemma 4.2. Let $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ be random variables with distributions $F_1,\,\ldots, \,F_n$ , $G_1,\,\ldots, \,G_n$ from class $\mathcal{L}$ , respectively. We also assume that $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ satisfy the GTAI property and it holds

\begin{eqnarray*} (X_k,\,Y_l)\in \mathcal{L}^{(2)}, \end{eqnarray*}

for any $k,\,l \in \{1,\,\ldots, \,n\}$ . Then it holds

\begin{eqnarray*} {\mathbf{P}}\left [S_n\gt x,\;T_n\gt y \right ] \gtrsim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ X_k\gt x,\; Y_l\gt y \right ], \end{eqnarray*}

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

Proof. We choose as $a(\cdot )$ a function with joint insensitivity property for any random pair $(X_k,\,Y_l)$ for any $k,\,l \in \{1,\,\ldots, \,n\}$ . A possible choice of this function is by (4.7). Next, we apply twice Bonferroni’s inequality to obtain

(4.8) \begin{eqnarray} &&{\mathbf{P}}\left [S_n\gt x,\;T_n\gt y \right ] \geq {\mathbf{P}}\left [S_n\gt x,\;T_n\gt y,\;\bigvee _{k=1}^n X_k\gt x+a(x),\; \bigvee _{l=1}^n Y_l\gt y+a(y) \right ]\nonumber \\ &&\geq \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [S_n\gt x,\;T_n\gt y,\; X_k\gt x+a(x),\; Y_l\gt y+a(y) \right ] \nonumber \\ &&- \sum \sum _{1\leq k \lt i \leq n} \sum _{l=1}^n {\mathbf{P}}\left [X_i\gt x+a(x),\; X_k\gt x+a(x),\; Y_l\gt y+a(y) \right ] \nonumber \\ &&- \sum _{k=1}^n \sum \sum _{1\leq l \lt i \leq n} {\mathbf{P}}\left [ X_k\gt x+a(x),\; Y_l\gt y+a(y),\;Y_i\gt y+a(y) \right ] \nonumber \\ &&\,=\!:\,\sum _{i=1}^3 J_i(x,\,y). \end{eqnarray}

Now for each term of $J_2(x,\,y)$ , we find

\begin{eqnarray*} && {\mathbf{P}}\left [X_i\gt x+a(x),\; X_k\gt x+a(x),\; Y_l\gt y+a(y) \right ] \\[2mm] &&={\mathbf{P}}\left [ X_i\gt x+a(x)\;|\; X_k\gt x+a(x),\;Y_l\gt y+a(y) \right ] \,{\mathbf{P}}\left [ X_k\gt x+a(x),\; Y_l\gt y+a(y) \right ] \\[2mm] &&=o({\mathbf{P}}\left [ X_k\gt x,\;Y_l\gt y \right ]), \end{eqnarray*}

as $x\wedge y \to \infty$ , which follows from GTAI property, $\mathcal{L}^{(2)}$ membership and the definition of function $a(\cdot )$ . So

(4.9) \begin{eqnarray} J_2(x,\,y) =o({\mathbf{P}}\left [ X_k\gt x,\;Y_l\gt y \right ]), \end{eqnarray}

as $x\wedge y \to \infty$ . Similarly, due to symmetry, we have

(4.10) \begin{eqnarray} J_3(x,\,y) =o({\mathbf{P}}\left [ X_k\gt x,\;Y_l\gt y \right ]), \end{eqnarray}

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

Finally, for the first term, we obtain

(4.11) \begin{eqnarray} J_1(x,\,y) &\geq & \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}[X_k\gt x+a(x),\;Y_l \gt y +a(y)]\nonumber \\ &-& \sum _{1\leq k \lt i \leq n}^n \sum \sum _{l=1}^n {\mathbf{P}}\left [X_k\gt x+a(x),\;Y_l \gt y +a(y),\; X_i\lt -\dfrac {a(x)}n\right ]\nonumber \\ &-& \sum _{k=1}^n \sum _{1\leq l \lt i \leq n}^n \sum {\mathbf{P}}\left [X_k\gt x+a(x),\;Y_l \gt y +a(y),\; Y_i\lt -\dfrac {a(y)}n\right ], \end{eqnarray}

hence, the last two terms in (4.11), from the $GTAI$ structure and the definition of the function $a$ , in combination with properties of class $\mathcal{L}^{(2)}$ , become negligible with respect to the first term in (4.11). Therefore, it holds

(4.12) \begin{eqnarray} J_1(x,\,y) &\gtrsim & \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}[X_k\gt x+a(x),\;Y_l \gt y +a(y)], \end{eqnarray}

as $x\wedge y \to \infty$ . Thus, relations (4.9), (4.10), and (4.12), together with relation (4.8), render the desired lower bound.

Lemma 4.3. Let $X_1,\,X_2,\,Y_1,\,Y_2$ be non-negative random variables, with $GTAI$ property, such that the pair $(X_k,\,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ , for any $k,\,l \in \{1,\,2\}$ . If by $a$ we denote the insensitivity function from (4.7) , then it holds

(4.13) \begin{eqnarray} {\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y),\; Y_2\leq \dfrac {y}2\right ] &\sim & {\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y),\; Y_2\leq \dfrac {y}2\right ]\nonumber \\ &\sim & {\mathbf{P}}[X_1\gt x,\;Y_1 \gt a(y)], \end{eqnarray}

as $x\wedge y \to \infty$ , and further, it holds

\begin{eqnarray*} {\mathbf{P}}\left [Y_1\gt y-a(y),\;X_1 \gt a(x),\; X_2\leq \dfrac {x}2\right ] &\sim & {\mathbf{P}}\left [Y_1\gt y,\;X_1 \gt a(x),\; X_2\leq \dfrac {x}2\right ]\\[2mm] &\sim & {\mathbf{P}}[Y_1\gt y,\;X_1 \gt a(x)], \end{eqnarray*}

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

Proof. We show only the first relation (4.13), since the second follows along similar steps, due to symmetry. At first, by definition of insensitivity function $a(\cdot )$ from (4.7), and if the pair $(X,\,Y) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ , we obtain

(4.14) \begin{eqnarray} 1\leq \lim _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}\left [X\gt x-a(x),\;Y \gt y\right ]}{{\mathbf{P}}\left [X\gt x,\;Y \gt y\right ]} \leq \lim _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}\left [X\gt x-a(x),\;Y \gt y-a(y)\right ]}{{\mathbf{P}}\left [X\gt x,\;Y \gt y\right ]}=1, \end{eqnarray}

Hence, because of $(X_1,\,Y_1) \in (\mathcal{D}\cap \mathcal{L})^{(2)} \subsetneq \mathcal{L}^{(2)}$ , we have through (4.14) that it holds

\begin{eqnarray*} {\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y) \right ]\sim {\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y)\right ], \end{eqnarray*}

or equivalently

(4.15) \begin{eqnarray} &&{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y),\;Y_2 \leq \dfrac y2 \right ] +{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y),\;Y_2 \gt \dfrac y2\right ]\nonumber \\ &&\sim {\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y),\;Y_2 \leq \dfrac y2 \right ] +{\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y),\;Y_2 \gt \dfrac y2\right ]\!, \end{eqnarray}

as $x\wedge y \to \infty$ . We compare the first terms of each side of (4.15), to find

\begin{eqnarray*} &&{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y),\;Y_2 \leq \dfrac y2 \right ]= {\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y)\right ]\\[2mm] &&-{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y),\;Y_2 \gt \dfrac y2 \right ] ={\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y)\right ]\\[2mm] &&-{\mathbf{P}}\left [Y_2\gt \dfrac y2 \;\Big |\;X_1\gt x-a(x),\;Y_1 \gt a(y) \right ] \,{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y)\right ]\\[2mm] &&\sim {\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y) \right ] -o\left ({\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y)\right ]\right )\!, \end{eqnarray*}

as $x\wedge y \to \infty$ , where in the pre-last step, we used the class $(\mathcal{D}\cap \mathcal{L})^{(2)}$ property, relation (4.14), and the $GTAI$ property. Similarly, we get

(4.16) \begin{eqnarray} &&{\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y),\;Y_2 \leq \dfrac y2 \right ]\nonumber \\ &&= {\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y)\right ]-{\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y),\,Y_2\gt \dfrac y2\right ]\nonumber \\ &&={\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y)\right ]-{\mathbf{P}}\left [Y_2\gt \dfrac y2 \;\Big |\;X_1\gt x,\;Y_1 \gt a(y)\right ]{\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y)\right ]\nonumber \\ &&\sim {\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y) \right ] -o\left ({\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y)\right ]\right ), \end{eqnarray}

as $x\wedge y \to \infty$ . Therefore, considering all together relations (4.15)–(4.16), we conclude relation (4.13).

Remark 4.1. Taking into account relations (4.15)–(4.16), together with the fact that the $X_1,\,X_2,\,Y_1,\,Y_2$ are non-negative random variables, which are $GTAI$ , it follows that

\begin{eqnarray*} {\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1 \gt a(y),\;Y_2 \gt \dfrac y2 \right ]= o({\mathbf{P}}\left [X_1\gt x,\;Y_1 \gt a(y) \right ] ),\\ {\mathbf{P}}\left [Y_1\gt y-a(y),\;X_1 \gt a(x),\;X_2 \gt \dfrac x2 \right ]= o({\mathbf{P}}\left [X_1\gt a(x),\;Y_1 \gt y \right ] ), \end{eqnarray*}

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

The next result shows that in the non-negative part of class $(\mathcal{D}\cap \mathcal{L})^{(2)}$ , the property of joint max-sum equivalence as also under an extra assumption the closure property with respect to convolution are satisfied, as soon as the GTAI holds.

Lemma 4.4. Let $X_1,\,X_2,\,Y_1,\,Y_2$ be non-negative random variables, with the following distributions $F_1,\,F_2,\,G_1,\,G_2$ from class $\mathcal{D}\cap \mathcal{L}$ , respectively. Further, we assume that the random variables $X_1,\,X_2,\,Y_1,\,Y_2$ satisfy the GTAI and

\begin{eqnarray*} (X_k,\,Y_l)\in (\mathcal{D}\cap \mathcal{L})^{(2)}, \end{eqnarray*}

for any $k,\,l \in \{1,\,2\}$ properties. Then it holds

(4.17) \begin{eqnarray} {\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ] \sim \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ], \end{eqnarray}

as $x\wedge y \to \infty$ . If further $X_1,X_2$ are TAI and $Y_1, Y_2$ are TAI, then

\begin{eqnarray*} (X_1+X_2,\,Y_1+Y_2) \in (\mathcal{D}\cap \mathcal{L})^{(2)}, \end{eqnarray*}

Proof. From Lemma4.2 and the fact that $(\mathcal{D}\cap \mathcal{L})^{(2)} \subsetneq \mathcal{L}^{(2)}$ , we find

(4.18) \begin{eqnarray} {\mathbf{P}}[X_1+X_2\gt x,\;Y_1+Y_2\gt y ]\gtrsim \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ], \end{eqnarray}

as $x\wedge y \to \infty$ , which provides the lower asymptotic bound.

Let us examine now the upper asymptotic bound

(4.19) \begin{eqnarray} &&{\mathbf{P}}[X_1+X_2\gt x,\;Y_1+Y_2\gt y ]\leq {\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1+Y_2\gt y \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_2\gt x-a(x),\;Y_1+Y_2\gt y \right ]+{\mathbf{P}}\left [X_1\gt a(x),\;X_2\gt \dfrac x2,\;Y_1+Y_2\gt y \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt \dfrac x2,\;X_2\gt a(x),\;Y_1+Y_2\gt y \right ] \leq {\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1\gt y-a(y) \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_2\gt y-a(y) \right ]+{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1\gt a(y),\;Y_2\gt \dfrac y2 \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt x-a(x),\;Y_1\gt \dfrac y2,\;Y_2\gt a(y) \right ]+{\mathbf{P}}\left [X_2\gt x-a(x),\;Y_1\gt y-a(y) \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_2\gt x-a(x),\;Y_2\gt y-a(y) \right ]+{\mathbf{P}}\left [X_2\gt x-a(x),\;Y_1\gt a(y),\;Y_2\gt \dfrac y2 \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_2\gt x-a(x),\;Y_1\gt \dfrac y2,\;Y_2\gt a(y) \right ]+{\mathbf{P}}\left [X_1\gt a(x),\;X_2\gt \dfrac x2,\;Y_1\gt y-a(y)\right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt a(x),\;X_2\gt \dfrac x2,\;Y_2\gt y-a(y) \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt a(x),\;X_2\gt \dfrac x2,\;Y_1\gt a(y),\;Y_2\gt \dfrac y2 \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt a(x),\;X_2\gt \dfrac x2,\;Y_1\gt \dfrac y2,\;Y_2\gt a(y) \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt \dfrac x2,\;X_2\gt a(x),\;Y_1\gt y-a(y)\right ]+{\mathbf{P}}\left [X_1\gt \dfrac x2,\;X_2\gt a(x),\;Y_2\gt y-a(y) \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt \dfrac x2,\;X_2\gt a(x),\;Y_1\gt a(y),\;Y_2\gt \dfrac y2 \right ]\nonumber \\ &&+{\mathbf{P}}\left [X_1\gt \dfrac x2, \;X_2\gt a(x),\;Y_1\gt \dfrac y2,\;Y_2\gt a(y) \right ]\,=\!:\,\sum _{i=1}^{16}I_i(x,\,y). \end{eqnarray}

Taking into account the property $\mathcal{L}^{(2)}$ and the definition of function $a(x)$ , we find the asymptotic expressions for $ I_1(x,\,y)\sim {\mathbf{P}}\left [X_1\gt x,Y_1\gt y\right ]$ , $I_2(x,\,y)\sim {\mathbf{P}}\left [X_1\gt x,Y_2\gt y\right ]$ , $I_5(x,\,y)\sim {\mathbf{P}}\left [X_2\gt x,Y_1\gt y\right ]$ , $I_6(x,\,y)\sim {\mathbf{P}}\left [X_2\gt x,Y_2\gt y\right ]$ , as $x\wedge y \to \infty$ . Hence

(4.20) \begin{eqnarray} I_1(x,\,y)+I_2(x,\,y)+I_5(x,\,y)+I_6(x,\,y) \sim \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ], \end{eqnarray}

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

Next, we follow a similar approach for $I_3(x,\,y)$ , $I_4(x,\,y)$ , $I_7(x,\,y)$ , $I_8(x,\,y)$ , $I_9(x,\,y)$ , $I_{10}(x,\,y)$ , $I_{13}(x,\,y)$ and $I_{14}(x,\,y)$ . Now, we obtain by Lemma4.3

\begin{eqnarray*} &&I_3(x,\,y) \sim {\mathbf{P}}\left [X_1\gt x,\;Y_1\gt a(y),\;Y_2\gt \dfrac y2 \right ]\\[2mm] &&\qquad ={\mathbf{P}}\left [Y_1\gt a(y)\;\Big |\;X_1\gt x,\;Y_2\gt \dfrac y2 \right ]\,{\mathbf{P}}\left [X_1\gt x,\;Y_2\gt \dfrac y2 \right ]=o\left ({\mathbf{P}}\left [X_2\gt x,Y_2\gt y\right ] \right ), \end{eqnarray*}

as $x\wedge y \to \infty$ , which follows because of properties $(\mathcal{D}\cap \mathcal{L})^{(2)}$ and GTAI.

In similar way, we find $I_4(x,\,y)=o({\mathbf{P}}\left [X_1\gt x,Y_2\gt y\right ])$ , $I_7(x,\,y)=o({\mathbf{P}}\left [X_2\gt x,Y_2\gt y\right ])$ , $I_8(x,\,y)=o({\mathbf{P}}\left [X_2\gt x,Y_2\gt y\right ])$ , $I_9(x,\,y)=o({\mathbf{P}}\left [X_2\gt x,Y_1\gt y\right ])$ and finally $I_{10}(x,\,y)=o({\mathbf{P}}\left [X_1\gt x,Y_2\gt y\right ])$ , as $x\wedge y \to \infty$ . Hence

(4.21) \begin{eqnarray} \sum _{j=3}^4 I_j(x,\,y)+ \sum _{i=7}^{10}I_i(x,\,y) +I_{13}(x,\,y) +I_{14}(x,\,y) =o\left (\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x, \,Y_l\gt y \right ]\right ), \end{eqnarray}

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

The $I_{11}(x,\,y)$ , $I_{12}(x,\,y)$ , $I_{15}(x,\,y)$ , $I_{16}(x,\,y)$ can be handled also similarly

\begin{eqnarray*} I_{11}(x,\,y) &\leq & {\mathbf{P}}\left [X_2\gt \dfrac x2,\;Y_1\gt a(y),\;Y_2\gt \dfrac y2 \right ]\\ &=&{\mathbf{P}}\left [Y_1\gt a(y)\;\Big |\;X_2\gt \dfrac x2,\;Y_2\gt \dfrac y2 \right ]\,{\mathbf{P}}\left [X_2\gt \dfrac x2, Y_2\gt \dfrac y2 \right ], \end{eqnarray*}

or equivalently $I_{11}(x,\,y) =o({\mathbf{P}}\left [X_2\gt x, Y_2\gt y \right ])$ , as $x\wedge y \to \infty$ , which follows because of properties $(\mathcal{D}\cap \mathcal{L})^{(2)}$ and GTAI. Similarly, we find $I_{1j}(x,\,y)=o({\mathbf{P}}\left [X_k\gt x, Y_l\gt y \right ])$ , for some $k,\,l \in \{1,\,2\}$ and for any $j\in \{1,\,2,\,5,\,6\}$ . Therefore, we obtain

(4.22) \begin{eqnarray} I_{1j}(x,\,y) =o\left (\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]\right ), \end{eqnarray}

as $x\wedge y \to \infty$ , for any $j\in \{1,\,2,\,5,\,6\}$ .

From (4.20), (4.21), and (4.22), in combination with (4.19), we find that

\begin{eqnarray*} {\mathbf{P}}[X_1+X_2\gt x,\;Y_1+Y_2\gt y ] \lesssim \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ] \end{eqnarray*}

as $x\wedge y \to \infty$ , which in combination with (4.18) leads to (4.17).

Now we check the validity of relation $(X_1+X_2,\,Y_1+Y_2) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ . At first, by (4.17), we obtain

\begin{eqnarray*} &&\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_1+X_2\gt b_1\,x,\;Y_1+Y_2\gt b_2\,y \right ]}{{\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ]}=\limsup _{x\wedge y \to \infty }\\[2mm] &&\dfrac { \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt b_1\,x,\;Y_l\gt b_2\,y \right ]}{ \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]} \leq \max _{k,\,l \in \{1, 2 \}} \left \{\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_k\gt b_1 x, \,Y_l\gt b_2 y \right ]}{{\mathbf{P}}\left [X_k\gt x,\,Y_l\gt y \right ]} \right \} \lt \infty, \end{eqnarray*}

for any $\textbf { b}=(b_1,\,b_2) \in (0,\,1)^2$ , this means that we have one of two conditions of the closure property with respect to $\mathcal{D}^{(2)}$ .

Next, we check the closure property with respect to $\mathcal{L}^{(2)}$ . From (4.17), we obtain

\begin{eqnarray*} &&\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_1+X_2\gt x-a_1,\;Y_1+Y_2\gt y-a_2 \right ]}{{\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ]}\\[2mm] &&\qquad \qquad =\limsup _{x\wedge y \to \infty }\dfrac {\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x-a_1,\;Y_l\gt y-a_2 \right ]}{\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]}, \end{eqnarray*}

for any $\textbf { a}=(a_1,\,a_2) \gt (0,\,0)$ , and therefore

\begin{eqnarray*} &&\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_1+X_2\gt x-a_1,\;Y_1+Y_2\gt y-a_2 \right ]}{{\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ]}\\[2mm] &&\qquad \qquad \leq \max _{k,\,l \in \{1,\,2 \}}\left \{ \limsup _{x\wedge y \to \infty }\dfrac { {\mathbf{P}}\left [X_k\gt x-a_1,\;Y_l\gt y-a_2 \right ]}{{\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]} \right \}=1, \end{eqnarray*}

and always

\begin{eqnarray*} \liminf _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_1+X_2\gt x-a_1,\;Y_1+Y_2\gt y-a_2 \right ]}{{\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ]}\geq 1, \end{eqnarray*}

which means that we have one of two conditions of the closure property with respect to $\mathcal{L}^{(2)}$ true. So by the extra assumption of TAI between $X_1,X_2$ and $Y_1,Y_2$ by Lemma 4.1 of Geluk and Tang (Reference Geluk and Tang2009), we have that $X_1+X_2\in \mathcal{D}\cap \mathcal{L}$ and $Y_1+Y_2\in \mathcal{D}\cap \mathcal{L}$ ; as a result, we conclude $(X_1+X_2,\,Y_1+Y_2) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ .

An easy example, where we combine the $GTAI$ property for the $X_1,\,\ldots, \,X_n$ , $Y_1,\,\ldots, \,Y_m$ with the $TAI$ property for each sequence, is found in case of each sequence to be $TAI$ but the two sequences to be independent.

Next, we provide a corollary, following from Lemma4.4, where we establish the closure property with respect to $\mathcal{C}^{(2)}$ and the joint max-sum equivalence, under condition GTAI.

Corollary 4.1. Let $X_1,\,X_2,\,Y_1,\,Y_2$ be non-negative random variables, with the distributions $F_1,\,F_2,\,G_1,\,G_2$ from class $\mathcal{C}$ , respectively, and they satisfy the GTAI condition. If it holds $(X_k,\,Y_l)\in \mathcal{C}^{(2)}$ , for any $k,\,l \in \{1,\,2\}$ , then

(4.23) \begin{eqnarray} {\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ]\sim \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ], \end{eqnarray}

as $x\wedge y \to \infty$ . If further $X_1,X_2$ are TAI and $Y_1, Y_2$ are TAI, then it holds $(X_1+X_2,\;Y_1+Y_2)\in \mathcal{C}^{(2)}$ .

Proof. Relation (4.23) follows from the fact that $\mathcal{C}^{(2)} \subsetneq (\mathcal{D}\cap \mathcal{L})^{(2)}$ (see Theorem2.1) and by application of Lemma4.4.

Next, we check the closure property with respect to convolution. From (4.23), we obtain

\begin{eqnarray*} {\mathbf{P}}[X_1+X_2\gt d_1\,x,\;Y_1+Y_2\gt d_2\,y ] \sim \sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt d_1\,x,\;Y_l\gt d_2\,y \right ], \end{eqnarray*}

as $x\wedge y \to \infty$ , for any $\textbf { d}=(d_1,\,d_2) \in (0,\,1)^2$ . Hence

\begin{eqnarray*} &&\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_1+X_2\gt d_1\,x,\;Y_1+Y_2\gt d_2\,y \right ]}{{\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ]}\\[2mm] &&\qquad \qquad =\limsup _{x\wedge y \to \infty }\dfrac {\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt d_1\,x,\;Y_l\gt d_2\,y \right ]}{\sum _{k=1}^2 \sum _{l=1}^2 {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]} \\[2mm] &&\qquad \qquad \leq \limsup _{x\wedge y \to \infty }\max _{k,\,l \in \{1,\,2 \}}\left \{ \dfrac { {\mathbf{P}}\left [X_k\gt d_1\,x,\;Y_l\gt d_2\,y \right ]}{{\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]} \right \}, \end{eqnarray*}

Thus, because of the definition of $\mathcal{C}^{(2)}$ , we get

\begin{eqnarray*} 1&\leq & \lim _{\textbf { d} \uparrow \textbf { 1}} \limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_1+X_2\gt d_1\,x,\;Y_1+Y_2\gt d_2\,y \right ]}{{\mathbf{P}}\left [X_1+X_2\gt x,\;Y_1+Y_2\gt y \right ]}\\[2mm] &\leq & \lim _{\textbf { d} \uparrow \textbf { 1}} \limsup _{x\wedge y \to \infty } \max _{k,\,l \in \{1,\,2\}} \left (\dfrac {{\mathbf{P}}\left [X_k\gt d_1\,x,\;Y_l\gt d_2\,y \right ]}{{\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]}\right )\\[2mm] &\leq & \max _{k,\,l \in \{1,\,2\}} \left (\lim _{\textbf { d} \uparrow \textbf { 1}} \limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_k\gt d_1\,x,\;Y_l\gt d_2\,y \right ]}{{\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ]}\right )=1, \end{eqnarray*}

this means that one of two conditions of closedness under convolution holds. By the assumptions of TAI in each sequence and by $\mathcal{C}\subsetneq \mathcal{D}\cap \mathcal{L}$ , we use Lemma 4.1 of Geluk and Tang (Reference Geluk and Tang2009), and we take

\begin{eqnarray*} 1&\leq & \lim _{{ d_1} \uparrow { 1}} \limsup _{x\to \infty }\dfrac {{\mathbf{P}}\left [X_1+X_2\gt d_1\,x \right ]}{{\mathbf{P}}\left [X_1+X_2\gt x \right ]}=\lim _{{ d_1} \uparrow { 1}} \limsup _{x \to \infty }\dfrac {{\mathbf{P}}\left [X_1\gt d_1\,x \right ]+{\mathbf{P}}\left [X_1\gt d_1\,x \right ]}{{\mathbf{P}}\left [X_1\gt x \right ]+{\mathbf{P}}\left [X_1\gt x \right ]}\\[2mm] &\leq & \lim _{{ d_1} \uparrow { 1}} \limsup _{x \to \infty } \max _{k \in \{1,\,2\}} \left (\dfrac {{\mathbf{P}}\left [X_k\gt d_1\,x \right ]}{{\mathbf{P}}\left [X_k\gt x \right ]}\right )\leq \max _{k \in \{1,\,2\}} \left (\lim _{{ d_1} \uparrow { 1}} \limsup _{x \to \infty }\dfrac {{\mathbf{P}}\left [X_k\gt d_1\,x \right ]}{{\mathbf{P}}\left [X_k\gt x \right ]}\right )=1, \end{eqnarray*}

which gives that $(X_1+X_2)\in \mathcal{C}$ . With the same argument, we have $(Y_1+Y_2)\in \mathcal{C}$ . That means $(X_1+X_2,\;Y_1+Y_2)\in \mathcal{C}^{(2)}$ .

Now we can give the main result, where we find an analogue to relation (4.2) in two dimensions.

Theorem 4.2. Let $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ be random variables with the following distributions $F_1,\,\ldots, \,F_n,\,G_1,\,\ldots, \,G_n$ from class $\mathcal{D}\cap \mathcal{L}$ , respectively, and they satisfy the GTAI condition, with $(X_k,\,Y_l)\in (\mathcal{D}\cap \mathcal{L})^{(2)}$ , for any $k,\,l \in \{1,\,\ldots, \,n\}$ . If further $X_1,\,\ldots, \,X_n$ are TAI and $Y_1,\,\ldots, \,Y_n$ are TAI, then

\begin{eqnarray*} {\mathbf{P}}\left [ \sum _{k=1}^n X_k\gt x,\;\sum _{l=1}^n Y_l \gt y\right ] &\sim & {\mathbf{P}}\left [ \bigvee _{i=1}^n S_i\gt x,\;\bigvee _{j=1}^n T_j \gt y\right ] \\[2mm] &\sim & {\mathbf{P}}\left [ \bigvee _{k=1}^n X_k\gt x,\;\bigvee _{l=1}^n Y_l \gt y\right ] \sim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [X_k\gt x,\;Y_l\gt y \right ], \end{eqnarray*}

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

Proof. By Lemma4.2, we find

\begin{eqnarray*} {\mathbf{P}}\left [\sum _{k=1}^n X_k\gt x,\;\sum _{l=1}^n Y_l\gt y \right ] \gtrsim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ X_k\gt x,\; Y_l\gt y \right ], \end{eqnarray*}

as $x\wedge y \to \infty$ . Because of closure property of $(\mathcal{D}\cap \mathcal{L})^{(2)}$ with respect to convolution in the positive part, under GTAI condition, we can apply Lemmas4.4 and 4.1, and employing induction, we find

\begin{eqnarray*} {\mathbf{P}}\left [\sum _{k=1}^n X_k\gt x, \sum _{l=1}^n Y_l\gt y \right ] \leq {\mathbf{P}}\left [\sum _{k=1}^n X_k^+\gt x, \sum _{l=1}^n Y_l^+\gt y \right ]\sim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [X_k\gt x, Y_l\gt y \right ] \end{eqnarray*}

as $x\wedge y \to \infty$ . Now, taking into consideration Theorem4.1, we find

\begin{eqnarray*} {\mathbf{P}}\left [\sum _{k=1}^n X_k\gt x,\;\sum _{l=1}^n Y_l\gt y \right ] \sim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ X_k\gt x,\; Y_l\gt y \right ] \sim {\mathbf{P}}\left [ \bigvee _{k=1}^n X_k\gt x,\;\bigvee _{l=1}^n Y_l \gt y\right ], \end{eqnarray*}

as $x\wedge y \to \infty$ . Finally, due to the inequality

\begin{eqnarray*} {\mathbf{P}}\left [\sum _{k=1}^n X_k\gt x, \sum _{l=1}^n Y_l\gt y \right ] \leq {\mathbf{P}}\left [ \bigvee _{i=1}^n S_i\gt x, \bigvee _{j=1}^n T_j \gt y\right ]\leq {\mathbf{P}}\left [\sum _{k=1}^n X_k^+\gt x, \sum _{l=1}^n Y_l^+\gt y \right ] \end{eqnarray*}

we get the asymptotic relation

\begin{eqnarray*} {\mathbf{P}}\left [ \bigvee _{i=1}^n S_i\gt x,\;\bigvee _{j=1}^n T_j \gt y\right ] \sim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ X_k\gt x,\; Y_l\gt y \right ], \end{eqnarray*}

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

Recently more and more researchers study two-dimensional risk models; we refer to the reader Hu and Jiang (Reference Hu and Jiang2013), Cheng and Yu (Reference Cheng and Yu2019), and Cheng (Reference Cheng2021), among others. For

\begin{eqnarray*} U_1(k,\,x)\,:\!=\,x - \sum _{i=1}^{k}X_i, \qquad U_2(k,\,y)\,:\!=\,y -\sum _{j=1}^{k}Y_j, \end{eqnarray*}

for $1\leq k\leq n$ , we define now two ruin times,

\begin{eqnarray*} T_{max}\,:\!=\,\inf \left \{1\leq k\leq n\;:\; U_1(k,\,x)\wedge U_2(k,\,y) \lt 0 \right \}\!, \end{eqnarray*}

which denote the first moment when both portfolios are found with negative surplus, and for each portfolio, we define

\begin{eqnarray*} T_{1}(x)\,:\!=\,\inf \left \{1\leq k\leq n: U_{1}(k,\,x)\lt 0 |U_{1}(0,\,x)=x\right \}\!, \end{eqnarray*}

\begin{eqnarray*} T_{2}(y)\,:\!=\,\inf \left \{1\leq k\leq n: U_{2}(k,\,y)\lt 0 |U_{2}(0,\,y)=y\right \}\!, \end{eqnarray*}

as a result, the second type of ruin type is

\begin{eqnarray*} T_{and}\,:\!=\,\max \left \{T_{1}(x),T_{2}(y)\right \}\!, \end{eqnarray*}

which corresponds to the first moment, when both portfolios have been with negative surplus, but not necessarily simultaneously. Hence we define the ruin probabilities as

(4.24) \begin{eqnarray} \psi _{max}(x,\,y,\,n) = {\mathbf{P}}[T_{max} \leq n],\qquad \psi _{and}(x,\,y,\,n) = {\mathbf{P}}[T_{and} \leq n]\!, \end{eqnarray}

for any $n \in {\mathbb{N}}$ and $x,\,y \gt 0$ . From (4.24), we easily find out that

\begin{eqnarray*} \psi _{and}(x,\,y,\,n) = {\mathbf{P}}\left [ \bigvee _{i=1}^n S_i \gt x,\;\bigvee _{i=1}^n T_i \gt y\right ]\!. \end{eqnarray*}

Therefore, by Theorem4.2, we obtain the following result.

Corollary 4.2. Under conditions of Theorem 4.2 , we obtain

(4.25) \begin{eqnarray} \psi _{and}(x,\,y,\,n) \sim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ X_k\gt x,\; Y_l\gt y \right ]\!, \end{eqnarray}

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

Remark 4.2. From relation (4.25) and the definitions for $T_{max}$ and $T_{and}$ , we can easily observe that $\psi _{max}(x,\,y,\,n) \leq \psi _{and}(x,\,y,\,n)$ , for any $x,\,y \gt 0$ and any $n \in {\mathbb{N}}$ . Thus, for $\psi _{max}(x,\,y,\,n)$ , we find the asymptotic upper bound

\begin{eqnarray*} \psi _{max}(x,\,y,\,n) \lesssim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ X_k\gt x,\; Y_l\gt y \right ]\!, \end{eqnarray*}

as $x\wedge y \to \infty$ , for any $n \in {\mathbb{N}}$ .

5. Scalar product

Now we examine the closure property of scalar product in $\mathcal{L}^{(2)}$ , $\mathcal{D}^{(2)}$ , and in their intersection. Later, we check the same for random sums in two dimensions.

The scalar product has the following tail:

(5.1) \begin{eqnarray} \boldsymbol { \overline {\mathbf{H}}} (x,\,y)\,:\!=\,{\mathbf{P}}[\Theta \,X \gt x,\,\Theta \,Y\gt y]. \end{eqnarray}

Here, we set $\Theta$ to be a non-negative random variable with distribution $B$ , such that $B(0-)=0$ and $B(0)\lt 1$ . We assume also that $\Theta$ is independent of $(X,\,Y)$ . These products in relation (5.1) have many applications in actuarial mathematics, in risk management, and in stochastic fields. Next, we use an assumption from Konstantinides and Passalidis (Reference Konstantinides and Passalidis2024b).

Remark 5.1. From Assumption 5.1 , it is implied that

(5.2) \begin{eqnarray} {\overline {B}}(c\,x) = o\left ({\mathbf{P}}\left [\Theta \,X\gt x\right ] \right ), \end{eqnarray}

as $x\to \infty$ , for any $c\gt 0$ , and similarly

(5.3) \begin{eqnarray} {\overline {B}}(c\,y) = o\left ({\mathbf{P}}\left [\Theta \,Y\gt y\right ] \right ), \end{eqnarray}

as $y\to \infty$ for any $c\gt 0$ . This condition is well-known; see, for example, Tang (Reference Tang2006). Further, we can see that Assumption 5.1 holds immediately when the distribution $B$ has support bounded from above (because of the unbounded support of $F,\,G$ ).

The next lemma helps our argumentation and presents a multivariate extension of Tang (Reference Tang2006, Lem. 3.2), providing the existence of an auxiliary function. For a similar paper on auxiliary functions, we refer to Zhou et al. (Reference Zhou, Wang and Wang2012).

Proof.

( $\Leftarrow$ ). The existence of such an auxiliary function easily implies Assumption5.1; for example, we consider the function $x\wedge y/n$ .

( $\Rightarrow$ ). Let suppose that Assumption5.1 is satisfied. Then we obtain

\begin{eqnarray*} \lim _{x\wedge y \to \infty } \dfrac {{\overline {B}}((x\wedge y) / n)}{{ \overline {H}}(x,\,y)}=0. \end{eqnarray*}

Let an increasing sequence of positive numbers $\{\lambda _n,\;n \in {\mathbb{N}}\}$ with $\lambda _{n+1}\gt (n+1)\,\lambda _n$ , for any $n \in {\mathbb{N}}$ , such that for any $x\wedge y \geq \lambda _n$ , we have

\begin{eqnarray*} \dfrac {{\overline {B}}(x\wedge y/n)}{{ \overline {H}}(x,\,y)} \leq \dfrac 1n . \end{eqnarray*}

Therefore, the points (1), (2), and (3) are satisfied with

\begin{eqnarray*} b(x\wedge y)\,:\!=\,\sup _{0\leq k \leq x\wedge y} z(k), \qquad \qquad z(x\wedge y) = \sum _{n=1}^{\infty } \dfrac {x\wedge y}n\,\textbf { 1}_{\{\lambda _n\leq x\wedge y \leq \lambda _{n+1} \}}, \end{eqnarray*}

which completes the proof.

Now we study the closure property of class $\mathcal{D}^{(2)}$ with respect to scalar product.

Proof. Initially, we get from $F,\,G \in \mathcal{D}$ and by Cline and Samorodnitsky (Reference Cline and Samorodnitsky1994, Th. 3.3 (i)) or Leipus, Šiaulys, and Konstantinides (2023, Prop. 5.4 (i)) that the products $\Theta \,X$ and $\Theta \,Y$ follow distributions from $\mathcal{D}$ . From Assumption5.1, we obtain that for any $\textbf { b} \in (0,\,1)^n$

(5.4) \begin{eqnarray} &&\limsup _{x\wedge y \to \infty }\dfrac {\boldsymbol { \overline {H}_b}(x,\,y)}{{ \overline {H}}(x,\,y)}=\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}[\Theta \,X\gt b_1\,x,\,\,\Theta \,Y\gt b_2\,y]}{{\mathbf{P}}[\Theta \,X\gt x,\;\Theta \,Y\gt y]}=\limsup _{x\wedge y \to \infty }\nonumber \\ &&\dfrac {\left (\int _0^{b(x\wedge y)}+\int _{b(x\wedge y)}^\infty \right ){\mathbf{P}}\left [X\gt \dfrac {b_1\,x}{s},\;Y\gt \dfrac {b_2\,y}s\right ]\,B(ds)}{{\mathbf{P}}[\Theta \, X\gt x,\;\Theta \, Y\gt y]}\,=\!:\,\limsup _{x\wedge y \to \infty }\dfrac {I_1+I_2}{{\mathbf{P}}[\Theta \, X\gt x,\Theta \, Y\gt y]}. \nonumber\\\end{eqnarray}

Further, we calculate

\begin{eqnarray*} I_2 \leq \int _{b(x\wedge y)}^\infty \,B(ds)={\overline {B}}[b(x\wedge y)] = o\left [{ \overline {H}}(x,\,y)\right ]\!, \end{eqnarray*}

as $x\wedge y \to \infty$ , due to Assumption5.1. Hence, taking into account also relation (5.4), we find

\begin{eqnarray*} \limsup _{x\wedge y \to \infty }\dfrac {\boldsymbol { \overline {H}_b}(x,\,y)}{{ \overline {H}}(x,\,y)}&\leq &\limsup _{x\wedge y \to \infty }\dfrac {\int _0^{b(x\wedge y)}{\mathbf{P}}\left [X\gt \dfrac {b_1\,x}{s},\;Y\gt \dfrac {b_2\,y}s\right ]\,B(ds)}{\int _0^{b(x\wedge y)}{\mathbf{P}}\left [X\gt \dfrac {x}{s},\;Y\gt \dfrac {y}s\right ]\,B(ds)}\\[2mm] &\leq &\limsup _{x\wedge y \to \infty } \sup _{0\lt s \leq b(x\wedge y)}\dfrac {{\mathbf{P}}\left [X\gt b_1\,\dfrac xs,\;Y\gt b_2\,\dfrac ys\right ]}{{\mathbf{P}}\left [X_1\gt \dfrac xs,\;Y\gt \dfrac ys\right ]}\\[2mm] &\leq &\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X\gt b_1\,x,\;Y\gt b_2\,y\right ]}{{\mathbf{P}}\left [X\gt x,\;Y\gt y\right ]}\lt \infty, \end{eqnarray*}

where in the last step, we used the condition $(X,\,Y) \in \mathcal{D}^{(2)}$ . So we get $\textbf { H}(x,\,y) \in \mathcal{D}^{(2)}$ .

Let us observe, that if $\Theta$ has upper bounded support, the proof of Theorem5.1 (as also of Theorem5.2) is implied by similar manipulations, replacing $b\,(x\wedge y)$ by the right endpoint of the distribution $B$ . Next, we provide an analogue for class $\mathcal{L}^{(2)}$ .

Proof. From the fact that $(X,\,Y)$ is independent of $\Theta$ , hold $F,\,G \in \mathcal{L}$ and relations (5.2) and (5.3), using Cline and Samorodnitsky (Reference Cline and Samorodnitsky1994, Th 2.2 (iii)), we find that distributions of $\Theta \,X$ and $\Theta \,Y$ belong to $\mathcal{L}$ . Let $\textbf { a}=(a_1,\,a_2) \gt (0,\,0)$ . Then we easily obtain

(5.5) \begin{eqnarray} \liminf _{x\wedge y \to \infty }\dfrac {{ \overline {H}} (x-a_1,\,y-a_2)}{{ \overline {H}} (x,\,y) }=\lim _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}[\Theta \,X \gt x-a_1,\,\Theta \,Y\gt y-a_2]}{{\mathbf{P}}[\Theta \,X \gt x,\;\Theta \,Y\gt y]} \geq 1. \end{eqnarray}

Next, we show the opposite asymptotic inequality. Using Assumption5.1, we obtain

(5.6) \begin{eqnarray} &&\limsup _{x\wedge y \to \infty }\dfrac {{ \overline {H}} (x-a_1,\,y-a_2)}{{ \overline {H}} (x,\,y) } \nonumber \\ &&=\lim _{x\wedge y \to \infty }\dfrac {1}{{ \overline {H}}(x,\,y)} \,\left (\int _0^{b(x\wedge y)} + \int _{b(x\wedge y)}^\infty \right ){\mathbf{P}}\left [ X\gt \dfrac {x -a_1}{s},\;Y\gt \dfrac {y-a_2}{s} \right ]B(ds)\\[2mm] \notag &&\,=\!:\,\lim _{x\wedge y \to \infty }\dfrac {I_1(x,\,y) +I_2(x,\,y) }{{ \overline {H}}(x,\,y) } . \end{eqnarray}

Thus, by Assumption5.1, we find

\begin{eqnarray*} I_2(x,\,y) = \int _{b(x\wedge y)}^\infty \,{\mathbf{P}}\left [ X\gt \dfrac {x -a_1}{s},\;Y\gt \dfrac {y-a_2}{s} \right ]\,B(ds) \leq {\overline {B}}[b(x\wedge y)] = o\left [ { \overline {H}} (x,\,y) \right ]\!, \end{eqnarray*}

hence,

\begin{eqnarray*} \dfrac {I_2(x,\,y)}{{ \overline {H}} (x,\,y)} = o(1), \end{eqnarray*}

as $x\wedge y \to \infty$ . As a consequence, taking into account also (5.6), we get

\begin{eqnarray*} \limsup _{x\wedge y \to \infty }\dfrac {\boldsymbol { \overline {H}_1} (x-a_1,\,y-a_2)}{{ \overline {H}} (x,\,y) }&=&\limsup _{x\wedge y \to \infty }\int _0^{b(x\wedge y)} {\mathbf{P}}\left [ X\gt \dfrac {x -a_1}{s},\;Y\gt \dfrac {y -a_2}{s} \right ]\dfrac {B(ds)}{{ \overline {H}} (x,\,y)} \\[2mm] &\leq & \limsup _{x\wedge y \to \infty }\dfrac {\int _0^{b(x\wedge y)} {\mathbf{P}}\left [ X\gt \dfrac {x -a_1}{s},\;Y\gt \dfrac {y -a_2}{s} \right ]B(ds)}{\int _0^{b(x\wedge y)} {\mathbf{P}}\left [ X\gt \dfrac { x }{s},\;Y\gt \dfrac {y }{s} \right ]B(ds)}\\[2mm] &\leq & \limsup _{x\wedge y \to \infty } \sup _{0\lt s \leq b(x\wedge y)}\dfrac {{\mathbf{P}}\left [ X\gt \dfrac {x -a_1}{s},\;Y\gt \dfrac {y-a_2}{s} \right ]}{{\mathbf{P}}\left [ X\gt \dfrac { x }{s},\;Y\gt \dfrac {y }{s} \right ]} \\[2mm] &=& \limsup _{x\wedge y \to \infty } \dfrac {{\mathbf{P}}\left [ X\gt x -a_1,\;Y\gt y -a_2 \right ]}{{\mathbf{P}}\left [ X\gt x,\;Y\gt y \right ]}=1 . \end{eqnarray*}

where in the last step, we consider the fact that $(X,\,Y) \in \mathcal{L}^{(2)}$ . So we have

(5.7) \begin{eqnarray} \limsup _{x\wedge y \to \infty }\dfrac {{ \overline {H}} (x-a_1,\,y-a_2)}{{ \overline {H}} (x,\,y) }\leq 1. \end{eqnarray}

From relations (5.5) and (5.7), we conclude $\textbf { H} (x,\,y) \in \mathcal{L}^{(2)}$ .

The next statement stems from a combination of previous results.

Proof. This follows directly from Theorems5.1 and 5.2.

6. Randomly weighted sums

Finally, we extend Theorem4.2 to weighted sums. The first kind of weighted sums takes the form

\begin{eqnarray*} S_n(\Theta )=\sum _{k=1}^n \Theta \,X_k, \qquad T_n(\Theta )=\sum _{l=1}^n \Theta \,Y_l. \end{eqnarray*}

These quantities have the same discount factor $\Theta$ ; hence, the $(X_k,\,Y_l)$ , for $k,\,l=1,\,\ldots, \,n$ , are the losses or gains of the two lines of business during the $k$ -th period. If the $(x,\,y)$ represents the two initial capitals, respectively, then the ruin probability in this model comes in the form

(6.1) \begin{eqnarray} \psi _{and}(x,\,y,\,n)\,:\!=\,{\mathbf{P}}\left [\bigvee _{i=1}^n S_i(\Theta )\gt x,\;\bigvee _{j=1}^n T_j(\Theta )\gt y \right ]. \end{eqnarray}

The ruin probability, in models with insurance and financial risks, plays a significant role in risk theory. For example, we refer to Li and Tang (Reference Li and Tang2015), Yang and Konstantinides (Reference Yang and Konstantinides2015), Cheng (Reference Cheng2021), and Ji et al. (Reference Ji, Wang, Yan and Cheng2023) for discrete-time or continuous-time models, respectively.

The next result is based on Theorem4.2 and Corollary5.1. We have to notice that there exists the asymptotic behavior of the ruin probability in (6.1) as well.

Corollary 6.1. Let $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ be random variables with the following distributions $F_1,\,\ldots, \,F_n,\,G_1,\,\ldots, \,G_n$ , respectively, from class $\mathcal{D}\cap \mathcal{L}$ , and they satisfy the GTAI dependence structure. We assume that $\Theta$ represents a non-negative random variable with upper bounded support, it is independent of $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ and $(X_k,\,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ , for $k,\,l=1,\,\ldots, \,n$ . If further $X_1,\,\ldots, \,X_n$ are TAI and $Y_1,\,\ldots, \,Y_n$ are TAI, then the following asymptotic relation is true:

(6.2) \begin{eqnarray} &&{\mathbf{P}}\left [ S_n(\Theta )\gt x,\; T_n(\Theta )\gt y \right ] \sim {\mathbf{P}}\left [\bigvee _{i=1}^n S_i(\Theta )\gt x,\;\bigvee _{j=1}^n T_j(\Theta )\gt y \right ] \nonumber \\ &&\sim {\mathbf{P}}\left [\bigvee _{k=1}^n \Theta \,X_k\gt x,\;\bigvee _{l=1}^n \Theta \,Y_l\gt y \right ] \sim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ \Theta \,X_k\gt x,\;\Theta \,Y_l\gt y \right ], \end{eqnarray}

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

Proof. We start from Konstantinides and Passalidis (Reference Konstantinides and Passalidis2024, Lem. 2.1), and because of the upper-bound of $\Theta$ , we obtain that the products $\Theta \,X_1,\,\ldots, \,\Theta \,X_n,\,\Theta \,Y_1,\,\ldots, \,\Theta \,Y_n$ are GTAI. Now we can apply Cline and Samorodnitsky (Reference Cline and Samorodnitsky1994, Th. 2.2 (iii), Th.3.3 (ii)) to find $\Theta \,X_k \in \mathcal{D}\cap \mathcal{L}$ , and $\Theta \,Y_l \in \mathcal{D}\cap \mathcal{L}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ . Because of the closedness of class $\mathcal{D}$ and using Theorem 2.2 of Li (Reference Li2013), we conclude that $\Theta _1\,X_1,\,\ldots, \,\Theta _n\,X_n$ are TAI and $\Delta _1\,Y_1,\,\ldots, \,\Delta _n\,Y_n$ are TAI.

Next, since $\Theta$ is bounded from above, so Assumption5.1 is fulfilled for $\Theta$ , $(X_k,\,Y_l)$ , for any $k,\,l=1,\,\ldots, \,n$ , in order to obtain $(\Theta \,X_k,\,\Theta \,Y_l) \in \mathcal{D}^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ , it is enough to apply Theorem5.1, and similarly, by Theorem5.2, we find $(\Theta \,X_k,\,\Theta \,Y_l) \in \mathcal{L}^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ . Therefore, the $(\Theta \,X_k,\,\Theta \,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ and the $\Theta \,X_1,\,\ldots, \,\Theta \,X_n$ , $\Theta \,Y_1,\,\ldots, \,\Theta \,Y_n$ are GTAI. Finally, applying Theorem4.2, we conclude (6.2).

Now we need some preliminary results. Several times before proving that the convolution product satisfies $H \in \mathcal{B}$ , with $\mathcal{B}$ some distribution class, we need to prove that $H_{\varepsilon }(x)\,:\!=\,{\mathbf{P}}[(\Theta \vee \varepsilon )\,X \leq x]$ belongs to this class $\mathcal{B}$ for any $\varepsilon \gt 0$ . Following the approach in Cline and Samorodnitsky (Reference Cline and Samorodnitsky1994), we show that for some constant $\delta \gt 0$ , if $H_{\varepsilon } \in \mathcal{L}^{(2)}$ , for any $\varepsilon \in (0,\,\delta )$ , then $H \in \mathcal{L}^{(2)}$ . However, the next results deserve theoretical attention by its own merit.

From here until the end of paper, we assume that $X,Y$ are non-negative random variables.

Proof. Keeping in mind that $((\Theta \vee \varepsilon _1)\,X,\,(\Delta \vee \varepsilon _2)\,Y) \in \mathcal{L}^{(2)}$ , we start by Cline and Samorodnitsky (Reference Cline and Samorodnitsky1994, th. 2.2 (i)) to establish that due to $(\Theta \vee \varepsilon _1)\,X \in \mathcal{L},\; (\Delta \vee \varepsilon _2)\,Y \in \mathcal{L}$ , we get $\Theta \,X \in \mathcal{L}$ and $\Delta \,Y\in \mathcal{L}$ . Next, we check the second property of class $\mathcal{L}^{(2)}$ . Let $(a_1,\,a_2) \gt (0,\,0)$ , then

(6.3) \begin{eqnarray} \liminf _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [\Theta \,X \gt x-a_1,\;\Delta \,Y\gt y -a_2 \right ]}{{\mathbf{P}}\left [\Theta \,X \gt x,\;\Delta \,Y\gt y \right ]} \geq 1, \end{eqnarray}

Next, for any $(\varepsilon _1,\,\varepsilon _2) \gt (0,\,0)$ , we find

\begin{eqnarray*} &&{\mathbf{P}}[(\Theta \vee \varepsilon _1)\,X\gt x,\;(\Delta \vee \varepsilon _2)\,Y\gt y] \geq {\mathbf{P}}[\Theta \,X\gt x,\;\Delta \,Y\gt y]\\[2mm] &&\geq {\mathbf{P}}[\Theta \,X\gt x,\;\Theta \gt \varepsilon _1,\;\Delta \,Y\gt y]\\[2mm] &&={\mathbf{P}}[(\Theta \vee \varepsilon _1)\,X\gt x,\;\Delta \,Y\gt y]-{\mathbf{P}}[\Theta \leq \varepsilon _1]\,{\mathbf{P}}[X\,\varepsilon _1\gt x,\,\Delta \,Y\gt y]\\[2mm] &&\geq {\mathbf{P}}[\Theta \gt \varepsilon _1]\,{\mathbf{P}}[(\Theta \vee \varepsilon _1)\,X\gt x,\,\Delta \,Y\gt y,\,\Delta \gt \varepsilon _2]=\\[2mm] &&{\mathbf{P}}[\Theta \gt \varepsilon _1] \left ({\mathbf{P}}[(\Theta \vee \varepsilon _1) X\gt x, (\Delta \vee \varepsilon _2) Y\gt y]-{\mathbf{P}}[(\Theta \vee \varepsilon _1) X\gt x,\, \varepsilon _2 Y\gt y] {\mathbf{P}}[\Delta \leq \varepsilon _2]\right )\\[2mm] &&\geq {\mathbf{P}}[\Theta \gt \varepsilon _1]\,{\mathbf{P}}[\Delta \gt \varepsilon _2]\,{\mathbf{P}}[(\Theta \vee \varepsilon _1)\,X\gt x,\,(\Delta \vee \varepsilon _2)\,Y\gt y], \end{eqnarray*}

hence we conclude

(6.4) \begin{eqnarray} &&{\mathbf{P}}[(\Theta \vee \varepsilon _1)\,X\gt x,\;(\Delta \vee \varepsilon _2)\,Y\gt y] \geq {\mathbf{P}}[\Theta \,X\gt x,\;\Delta \,Y\gt y]\nonumber \\ &&\geq {\mathbf{P}}[\Theta \gt \varepsilon _1]\,{\mathbf{P}}[\Delta \gt \varepsilon _2]\,{\mathbf{P}}[(\Theta \vee \varepsilon _1)\,X\gt x,\,(\Delta \vee \varepsilon _2)\,Y\gt y]. \end{eqnarray}

Therefore, using (6.4) and due to properties of $\mathcal{L}^{(2)}$ , for $((\Theta \vee \varepsilon _1)\,X,\,(\Delta \vee \varepsilon _2)\,Y)$ , we obtain

\begin{eqnarray*} &&\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [\Theta \,X \gt x-a_1,\;\Delta \,Y\gt y -a_2 \right ]}{{\mathbf{P}}\left [\Theta \,X \gt x,\;\Delta \,Y\gt y \right ]} \leq \\[2mm] &&\limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [(\Theta \vee \varepsilon _1)\,X \gt x-a_1,\;(\Delta \vee \varepsilon _2)\,Y\gt y -a_2 \right ]}{{\mathbf{P}}[\Theta \gt \varepsilon _1] {\mathbf{P}}[\Delta \gt \varepsilon _2] {\mathbf{P}}\left [(\Theta \vee \varepsilon _1) X \gt x, (\Delta \vee \varepsilon _2) Y\gt y\right ]}=\dfrac {1}{{\mathbf{P}}[\Theta \gt \varepsilon _1] {\mathbf{P}}[\Delta \gt \varepsilon _2]}, \end{eqnarray*}

and leaving $\varepsilon _1$ and $\varepsilon _2$ to tend to zero, we get

\begin{eqnarray*} \limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [\Theta \,X \gt x-a_1,\;\Delta \,Y\gt y -a_2 \right ]}{{\mathbf{P}}\left [\Theta \,X \gt x,\;\Delta \,Y\gt y \right ]} \leq 1, \end{eqnarray*}

hence, from (6.3) and from last inequality, we reach to $(\Theta \,X,\,\Delta \,Y) \in \mathcal{L}^{(2)}$ .

Lemma 6.2. Let $X$ and $Y$ be non-negative random variables, with $(X,\,Y) \in \mathcal{L}^{(2)}$ and $\Theta$ and $\Delta$ be non-negative, non-degenerated to zero random variables, independent of $(X,\,Y)$ . We assume that

(6.5) \begin{eqnarray} &&{\mathbf{P}}\left [\Theta \gt x\right ]=o({\mathbf{P}}\left [\Theta \,X \gt c_1\,x,\;\Delta \,Y\gt c_2\,y \right ])= {\mathbf{P}}\left [\Delta \gt y\right ], \end{eqnarray}

as $x \wedge y \to \infty$ , for any $c_1,\,c_2 \gt 0$ . Then $(\Theta \,X,\,\Delta \,Y) \in \mathcal{L}^{(2)}$ .

Proof. From (6.5), we obtain

(6.6) \begin{eqnarray} \dfrac {{\mathbf{P}}\left [\Theta \gt x \right ]}{{\mathbf{P}}\left [\Theta \,X \gt c_1\,x\right ]} \leq \dfrac {{\mathbf{P}}\left [\Theta \gt x \right ]}{{\mathbf{P}}\left [\Theta \,X \gt c_1\,x,\;\Delta \,Y\gt c_2\,y\right ]} \longrightarrow 0, \end{eqnarray}

as $x\wedge y \to \infty$ , and similarly, we find ${\mathbf{P}}\left [\Delta \gt y\right ]=o({\mathbf{P}}\left [\Delta \,Y\gt c_2\,y \right ])$ , as $x\wedge y \to \infty$ , for any $c_1,\,c_2 \gt 0$ . Hence, by Cline and Samorodnitsky (Reference Cline and Samorodnitsky1994,Th. 2.2), we find $\Theta \,X \in \mathcal{L}$ and $\Delta \,Y \in \mathcal{L}$ . Next, we show the second property of $(\Theta \,X,\,\Delta \,Y) \in \mathcal{L}^{(2)}$ . Indeed, from Lemma6.1, we see that it is enough to show this for any $\Theta \geq \varepsilon _1$ and $\Delta \geq \varepsilon _2$ almost surely for any $\varepsilon _1,\,\varepsilon _2 \gt 0$ . Let consider some $a_1,\,a_2 \gt 0$ and some $k_1,\,k_2,\,k \gt 0$ , such that for a large enough $x_0 \geq 0$ , it holds

(6.7) \begin{eqnarray} {\mathbf{P}}\left [X \gt x-\dfrac {a_1}{\varepsilon _1},\;Y\gt y -\dfrac {a_2}{\varepsilon _2} \right ]\leq (1+k)\,{\mathbf{P}}\left [X \gt x,\;Y\gt y \right ], \end{eqnarray}

for any $x \wedge y \geq x_0$ and

(6.8) \begin{eqnarray} {\mathbf{P}}\left [X \gt x-\dfrac {a_1}{\varepsilon _1}\right ]\leq (1+k_1)\,{\mathbf{P}}\left [X \gt x \right ],\quad {\mathbf{P}}\left [Y\gt y -\dfrac {a_2}{\varepsilon _2} \right ]\leq (1+k_2)\,{\mathbf{P}}\left [Y\gt y \right ], \end{eqnarray}

for any $x \geq x_0$ and $y \geq x_0$ , respectively. Then, for all $x \wedge y \geq x_0$ , we obtain

(6.9) \begin{eqnarray} &&{\mathbf{P}}\left [\Theta \,X \gt x-a_1,\;\Delta \,Y\gt y -a_2 \right ]=\left (\int _{\varepsilon _1}^{x/x_0}+\int _{x/x_0}^\infty \right )\,\nonumber \\ &&\left (\int _{\varepsilon _2}^{y/x_0}+\int _{y/x_0}^\infty \right )\, {\mathbf{P}}\left [X \gt \dfrac {x-a_1}{s},\;Y\gt \dfrac {y-a_2}{t} \right ] \,{\mathbf{P}}[\Theta \in ds,\;\Delta \in dt]\,=\!:\,\sum _{m=1}^4 I_m(x,\,y), \end{eqnarray}

where we find

\begin{eqnarray*} I_4(x,\,y) = \int _{x/x_0}^\infty \,\int _{y/x_0}^\infty {\mathbf{P}}\left [X \gt \dfrac {x-a_1}{s},\;Y\gt \dfrac {y-a_2}{t} \right ] \,{\mathbf{P}}[\Theta \in ds,\;\Delta \in dt], \end{eqnarray*}

that gives

(6.10) \begin{eqnarray} I_4(x,\,y) \leq {\mathbf{P}}\left [\Theta \geq \dfrac x{x_0},\;\Delta \geq \dfrac y{x_0}\right ]. \end{eqnarray}

Now we estimate $I_1(x,\,y)$

(6.11) \begin{eqnarray} I_1(x,\,y) &=& \int _{\varepsilon _1}^{x/x_0}\,\int _{\varepsilon _2}^{y/x_0} {\mathbf{P}}\left [X \gt \dfrac {x-a_1}{s},\;Y\gt \dfrac {y-a_2}{t} \right ] \,{\mathbf{P}}[\Theta \in ds,\;\Delta \in dt]\\[2mm] \notag &\leq & \int _{\varepsilon _1}^{x/x_0}\,\int _{\varepsilon _2}^{y/x_0}\,{\mathbf{P}}\left [X \gt \dfrac {x}{s}-\dfrac {a_1}{\varepsilon _1},\;Y\gt \dfrac {y}{t}-\dfrac {a_2}{\varepsilon _2} \right ] \,{\mathbf{P}}[\Theta \in ds,\;\Delta \in dt]\\[2mm] &\leq & (1+k)\,\int _{\varepsilon _1}^{x/x_0}\,\int _{\varepsilon _2}^{y/x_0}\,{\mathbf{P}}\left [X \gt \dfrac {x}{s},\;Y\gt \dfrac {y}{t} \right ] \,{\mathbf{P}}[\Theta \in ds,\;\Delta \in dt]\nonumber \\ &\leq & (1+k)\,{\mathbf{P}}\left [\Theta \,X \gt x,\;\Delta \,Y\gt y \right ], \nonumber \end{eqnarray}

thus we get $I_1(x,\,y) \leq (1+k)\,{\mathbf{P}}\left [\Theta \,X\gt x,\;\Delta \,Y \gt y\right ]$ , which follows from (6.7).

Next we consider $I_2(x,\,y)$

(6.12) \begin{eqnarray} I_2(x,\,y) &=& \int _{\varepsilon _1}^{x/x_0}\,\int _{y/x_0}^\infty {\mathbf{P}}\left [X \gt \dfrac {x-a_1}{s},\;Y\gt \dfrac {y-a_2}{t} \right ] \,{\mathbf{P}}[\Theta \in ds,\;\Delta \in dt]\\[2mm] \notag &\leq & \int _{\varepsilon _1}^{x/x_0}\,{\mathbf{P}}\left [X \gt \dfrac {x}{s}-\dfrac {a_1}{\varepsilon _1} \right ] \,{\mathbf{P}}\left [\Theta \in ds,\;\Delta \gt \dfrac y{x_0} \right ]\\[2mm] &\leq & (1+k_1)\, \int _{\varepsilon _1}^{x/x_0}\,{\mathbf{P}}\left [X \gt \dfrac {x}{s} \right ] \,{\mathbf{P}}\left [\Theta \in ds,\;\Delta \gt \dfrac y{x_0} \right ]\nonumber \\ &\leq & (1+k_1)\,{\mathbf{P}}\left [\Theta \,X \gt x,\;\Delta \gt \dfrac y{x_0} \right ] \leq (1+k_1)\,{\mathbf{P}}\left [\Delta \gt \dfrac y{x_0} \right ], \nonumber\end{eqnarray}

which means $I_2(x,\,y) \leq (1+k_1)\,{\mathbf{P}}\left [\Delta \gt y/{x_0} \right ]$ , where in the pre-last step, we use the first relation in (6.8).

For $I_3(x,\,y)$ , we use the second relation in (6.8), and due to symmetry with respect to (6.13), we find

(6.13) \begin{eqnarray} I_3(x,\,y) &=&\int _{x/x_0}^{\infty } \int _{\varepsilon _2}^{\infty }\, {\mathbf{P}}\left [X \gt \dfrac {x-a_1}{s},\;Y\gt \dfrac {y-a_2}{t} \right ] \,{\mathbf{P}}[\Theta \in ds,\;\Delta \in dt]\nonumber \\ &\leq & (1+k_2)\,{\mathbf{P}}\left [\Theta \gt \dfrac x{x_0} \right ]. \end{eqnarray}

Therefore, putting the estimation from (6.10) to (6.15) into (6.9), we conclude

\begin{eqnarray*} &&{\mathbf{P}}\left [\Theta \,X \gt x-a_1,\;\Delta \,Y\gt y -a_2 \right ] \leq {\mathbf{P}}\left [\Theta \gt \dfrac x{x_0},\;\Delta \gt \dfrac y{x_0} \right ] \\ &&\qquad + (1+k)\,{\mathbf{P}}\left [\Theta \,X\gt x,\;\Delta \,Y\gt y \right ]+ (1+k_2)\,{\mathbf{P}}\left [\Theta \gt \dfrac x{x_0} \right ]+ (1+k_1)\,{\mathbf{P}}\left [\Delta \gt \dfrac y{x_0} \right ], \end{eqnarray*}

Now, because of (6.5) and the relation

\begin{eqnarray*} \dfrac {{\mathbf{P}}\left [\Theta \gt x,\; \Delta \gt y\right ]}{{\mathbf{P}}\left [\Theta \,X \gt c_1\,x,\;\Delta \,Y\gt c_2\,y\right ]} \leq \dfrac {{\mathbf{P}}\left [\Theta \gt x \right ]}{{\mathbf{P}}\left [\Theta \,X \gt c_1\,x,\;\Delta \,Y\gt c_2\,y\right ]} \longrightarrow 0, \end{eqnarray*}

as $x \wedge y \to \infty$ , for any $c_1,\,c_2 \gt 0$ , we find

\begin{eqnarray*} \lim _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [\Theta \,X \gt x-a_1,\;\Delta \,Y\gt y -a_2 \right ]}{{\mathbf{P}}\left [\Theta \,X \gt x,\;\Delta \,Y\gt y \right ]} \leq 1+k. \end{eqnarray*}

By these inequalities and relation (6.6), in combination of the arbitrary choice of $k$ and relation (6.3), we have $(\Theta \,X,\,\Delta \,Y) \in \mathcal{L}^{(2)}$ .

Next, we consider the asymptotic joint-tail behavior of discounted aggregate claims in a two-dimensional discrete-time risk model, where the vector $(X_k,\,Y_k)$ represents losses in two lines of business at the $k$ -th period, while the $(\Theta _k,\,\Delta _k)$ represents the discount factors of these two lines of business, respectively. In this risk model, we study only the aggregate claims, and we accept that the $\Theta _1,\,\ldots, \,\Theta _n,\,\Delta _1,\,\ldots, \,\Delta _m$ are independent of claims $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_m$ . For further reading on risk models with dependence among the discount factors and main claims, see Chen (Reference Chen2011, Reference Chen2017) and Yang et al. (Reference Yang, Gao and Li2016), but only in one dimension. Specifically, we have the sums:

\begin{eqnarray*} S_{n}^{\Theta }\,:\!=\,\sum _{k=1}^{n}\Theta _k X_k,\qquad T_{n}^{\Delta }\,:\!=\,\sum _{l=1}^{n}\Delta _l Y_l. \end{eqnarray*}

Theorem 6.1. Let $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ be non-negative, random variables with the following distributions $F_1,\,\ldots, \,F_n,\,G_1,\,\ldots, \,G_n$ , respectively, from class $\mathcal{D}\cap \mathcal{L}$ , and they satisfy the GTAI dependence structure, with $(X_k,\,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ . We suppose that the random discount factors $\Theta _1,\,\ldots, \,\Theta _n,\,\Delta _1,\,\ldots, \,\Delta _n$ satisfy Assumption 6.1 and are independent of $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ . Then the products $\Theta _1\,X_1,\,\ldots, \,\Theta _n\,X_n$ , $\Delta _1\,Y_1,\,\ldots, \,\Delta _n\,Y_n$ are GTAI with $(\Theta _k\,X_k,\,\Delta _l\,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ . If further $X_1,\,\ldots, \,X_n$ are TAI and $Y_1,\,\ldots, \,Y_n$ are TAI, then the following asymptotic relations hold

(6.14) \begin{align} {\mathbf{P}}\left [S_n^{\Theta }\gt x,\;T_n^{\Delta }\gt y\right ] &\sim {\mathbf{P}}\left [\bigvee _{i=1}^n S_i^{\Theta }\gt x,\;\bigvee _{j=1}^n T_j^{\Delta }\gt y\right ] \nonumber \\ &\sim \sum _{k=1}^n \sum _{l=1}^n {\mathbf{P}}\left [ \Theta \,X_k\gt x,\;\Delta \,Y_l\gt y \right ], \end{align}

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

Proof. Taking into account the upper bound for discount factors $\Theta _1,\,\ldots, \,\Theta _n,\,\Delta _1,\,\ldots, \,\Delta _n$ and their independence from $X_1,\,\ldots, \,X_n,\,Y_1,\,\ldots, \,Y_n$ , we apply Konstantinides and Passalidis (Reference Konstantinides and Passalidis2024, Lem. 2.1) to find that the products $\Theta _1\,X_1,\,\ldots, \,\Theta _n\,X_n,\,\Delta _1\,Y_1,\,\ldots, \,\Delta _n\,Y_n$ are GTAI. Now by Cline and Samorodnitsky (Reference Cline and Samorodnitsky1994, Th. 3.3 (i)), we get $\Theta _k\,X_k \in \mathcal{D}\cap \mathcal{L}$ and $\Delta _l\,Y_l \in \mathcal{D}\cap \mathcal{L}$ , for any $k=1,\,\ldots, \,n$ and for any $l=1,\,\ldots, \,n$ . As a result, by class $\mathcal{D}$ , using Li (Reference Li2013, Th. 2.2), the $\Theta _1\,X_1,\,\ldots, \,\Theta _n\,X_n$ are TAI, and $\Delta _1\,Y_1,\,\ldots, \,\Delta _n\,Y_n$ are TAI.

Next, we check if $(\Theta _k\,X_k,\,\Delta _l\,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ . Let $\textbf {b}=(b_1,\,b_2) \in (0,\,1)^2$ , then

\begin{eqnarray*} \limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [\Theta _k\,X_k \gt b_1\,x,\;\Delta _l\,Y_l\gt b_2\,y \right ]}{{\mathbf{P}}\left [\Theta _k\,X_k \gt x,\,\Delta _l\,Y_l\gt y \right ]}\leq \limsup _{x\wedge y \to \infty }\dfrac {{\mathbf{P}}\left [X_k \gt b_1\dfrac x{\delta _k},\,Y_l\gt b_2\dfrac y{\zeta _l} \right ]}{{\mathbf{P}}\left [X_k \gt \dfrac x{\xi _k},\;Y_l\gt \dfrac y{\gamma _l} \right ]} \lt \infty, \end{eqnarray*}

which follows from the inequalities

\begin{eqnarray*} \dfrac {b_1}{\delta _k} \lt \dfrac 1{\xi _k}, \qquad \dfrac {b_2}{\zeta _l} \lt \dfrac 1{\gamma _l}, \end{eqnarray*}

and the membership $(X_k,\,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ . Hence, we find the relation $(\Theta _k\,X_k,\,\Delta _l\,Y_l) \in \mathcal{D}^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ .

Now, noticing that relation (6.5) is satisfied because of Assumption6.1, we obtain directly from Lemma6.2 the inclusion $(\Theta _k\,X_k,\,\Delta _l\,Y_l) \in \mathcal{L}^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ . Hence $(\Theta _k\,X_k,\,\Delta _l\,Y_l) \in (\mathcal{D}\cap \mathcal{L})^{(2)}$ for any $k=1,\,\ldots, \,n$ and $l=1,\,\ldots, \,n$ and by application of Theorem4.2 for the products, we conclude relation (6.16).