2. Preliminaries

In what follows, a distribution $V=1-\overline{V}$ is always assumed to have an infinite upper endpoint, i.e., $\overline{V}(x)>0$ for any $x\in ({-}\infty,\infty )$ . In extreme value theory, V is said to belong to the Fréchet MDA if there is some $\alpha \geq 0$ such that the relation

(2.1) \begin{equation}\lim_{t\rightarrow \infty }\frac{\overline{V}(tx)}{\overline{V}(t)}=x^{-\alpha } \end{equation}

holds for any $x>0$ . In this case, V is also said to be from the class of regular variation, and we express the regularity property in (2.1) as $V\in \mathcal{R}_{-\alpha }$ , so that $\mathcal{R}$ is the union of all $\mathcal{R}_{-\alpha }$ over the range $\alpha\geq 0$ . The class $\mathcal{R}$ is an important class of heavy-tailed distributions, and its main members include the Pareto distribution, Student’s t-distribution, and the log-gamma distribution. See Bingham et al. [Reference Bingham, Goldie and Teugels7] for a monograph on regular variation. By definition, V is said to belong to the Gumbel MDA (with an infinite upper endpoint) if there is some positive auxiliary function h such that the relation

(2.2) \begin{equation}\lim_{t\rightarrow \infty }\frac{\overline{V}(t+h(t)x)}{\overline{V}(t)}=\textrm{e}^{-x} \end{equation}

holds for any $x\in ({-}\infty ,\infty )$ . We denote by $V\in\textrm{GMDA}(h)$ the property stated in (2.2). It is known that the function h is unique up to asymptotic equivalence and satisfies $h(t)=o(t)$ ; see Chapter 1.1 of Resnick [Reference Resnick40] or Chapter 3.3.3 of Embrechts et al. [Reference Embrechts, Klüppelberg and Mikosch18]. The Gumbel MDA contains both light-tailed distributions (e.g., the exponential and normal distributions) and heavy-tailed distributions (e.g., the log-normal distribution). It is easy to check that if $V\in \textrm{GMDA}(h)$ , then V belongs to the class of rapid variation, which is denoted by $\mathcal{R}_{-\infty }$ and is characterized by the following relation:

\begin{equation*}\lim_{t\rightarrow \infty}\frac{\overline{V}(tx)}{\overline{V}(t)}=0,\quad x>1.\end{equation*}

Clearly, the concepts of regular and rapid variation can be naturally extended to a general positive function g. Namely, for some $\beta \in[{-}\infty ,\infty ]$ , we write $g\in \mathcal{R}_{\beta }$ if (2.1) holds with $\overline{V}$ and $-\alpha $ replaced by g and $\beta $ , respectively. The well-known Karamata-type results hold for regularly and rapidly varying functions; i.e., if $g\in\mathcal{R}_{\beta }$ with $\beta \in ({-}\infty ,-1)$ , then

(2.3) \begin{equation}\lim_{t\rightarrow \infty }\frac{\int_{t}^{\infty }g(x)\textrm{d}x}{tg(t)}=-\frac{1}{\beta +1}, \end{equation}

while if $\beta =-\infty $ and g is non-increasing, then it holds for any $r\in ({-}\infty ,\infty )$ that

(2.4) \begin{equation}\lim_{t\rightarrow \infty }\frac{\int_{t}^{\infty }x^{r}g(x)\textrm{d}x}{t^{r+1}g(t)}=0. \end{equation}

See, e.g., Appendix A3 of Embrechts et al. [Reference Embrechts, Klüppelberg and Mikosch18] for more details.

Given d real-valued random variables $Z_{1}$ , …, $Z_{d}$ without upper bounds, we say they are pairwise asymptotically independent if, for each pair $1\leq i\neq j\leq d$ ,

(2.5) \begin{equation}\lim_{t\rightarrow \infty }\frac{\mathbb{P}(\!\left\vert Z_{i}\right\vert>t,Z_{j}>t)}{\mathbb{P}(Z_{i}>t)+\mathbb{P}(Z_{j}>t)}=0; \end{equation}

see, among many others, Chen and Yuen [Reference Chen and Yuen12], Li [Reference Li33], and Leipus et al. [Reference Leipus, Paukštys and Šiaulys32] for discussions and applications of this dependence structure. Note that the relation (2.5) will play an important role in dealing with joint probabilities related to the left or right tails of the random variables. If the right tails of $Z_{1}$ , …, $Z_{d}$ are asymptotically proportionally equivalent, i.e., $\lim_{t\rightarrow \infty }\mathbb{P}(Z_{i}>t)/\mathbb{P}(Z_{1}>t)=c_{i}$ for each $1\leq i\leq d$ and some $c_{i}>0$ , then the relation (2.5) is equivalent to

(2.6) \begin{equation}\lim_{t\rightarrow \infty }\frac{\mathbb{P}(\!\left\vert Z_{i}\right\vert>t,Z_{j}>t)}{\mathbb{P}(Z_{1}>t)}=0. \end{equation}

If, further, $Z_{1}$ has a regularly varying tail, then (2.6) implies that for any $a>0$ and $b>0$ ,

(2.7) \begin{equation}\lim_{x\rightarrow \infty }\frac{\mathbb{P}(\!\left\vert Z_{i}\right\vert>at,Z_{j}>bt)}{\mathbb{P}(Z_{1}>t)}=0. \end{equation}

To verify (2.7), we only need to note that

\begin{equation*}\limsup_{t\rightarrow \infty }\frac{\mathbb{P}(\!\left\vert Z_{i}\right\vert>at,Z_{j}>bt)}{\mathbb{P}(Z_{1}>t)}\leq \lim_{t\rightarrow \infty }\frac{\mathbb{P}\!\left( \left\vert Z_{i}\right\vert >\min\{a,b\}t,X_{j}>\min\{a,b\}t\right) }{\mathbb{P}\!\left( Z_{1}>\min \{a,b\}t\right) }\frac{\mathbb{P}(Z_{1}>\min \{a,b\}t)}{\mathbb{P}(Z_{1}>t)}=0.\end{equation*}

Hereafter, unless otherwise stated, all limit relationships hold as $t\rightarrow \infty $ . For two positive functions $g_{1}$ and $g_{2}$ , we write $g_{1}\left( t\right) \lesssim g_{2}\left(t\right) $ or $g_{2}\left( t\right) \gtrsim g_{1}\left( t\right) $ if $\limsup g_{1}\left( t\right) /g_{2}\left( t\right) \leq 1$ ; we write $g_{1}\left( t\right) \sim g_{2}\left( t\right) $ if $\lim g_{1}\left( t\right) /g_{2}\left( t\right) =1$ ; and we write $g_{1}\left( t\right) \asymp g_{2}\left( t\right) $ if $0\lt \lim\, \textrm{inf}\,g_{1}\left( t\right) /g_{2}\left( t\right) \leq \limsup g_{1}\left(t\right) /g_{2}\left( t\right) \lt \infty $ . For a real number a, we write $a^{+}=\max \{a,0\}$ and $a^{-}=-\min \{a,0\}$ . As usual, $\textbf{1}_{\{\cdot \}}$ stands for the indicator function.

3. Main results

In this section, we present our main asymptotic results for the TM and TCM defined by (1.1) and (1.2) with $n\in\textbf{N}^{+}$ . Denote by $F_{1}$ , …, $F_{d}$ the distributions of the individual risks $X_{1} $ , …, $X_{d}$ . We conduct our study under the following two assumptions, respectively.

Assumption 3.2. $F_{1}\in \textrm{GMDA}(h)$ , and $\overline{F}_{i}(t)\sim c_{i}\overline{F}_{1}(t)$ and $F_{i}({-}t)=O(\overline{F}_{1}(t))$ for each $1\leq i\leq d$ and some $c_{i}>0$ . Also, for each pair $1\leq i\neq j\leq d$ ,

(3.1) \begin{equation}\lim_{t\rightarrow \infty }\frac{\mathbb{P}\!\left( \left\vert X_{i}\right\vert >\epsilon h(t),X_{j}>t\right) }{\overline{F}_{1}(t)}=0,\quad {for\ any}\;\;\epsilon >0, \end{equation}

and

(3.2) \begin{equation}\lim_{t\rightarrow \infty }\frac{\mathbb{P}\!\left(X_{i}>Lh(t),X_{j}>Lh(t)\right) }{\overline{F}_{1}(t)}=0,\quad {for\ some}\;\;L>0. \end{equation}

The conditions regarding the marginal distributions of $(X_{1},\ldots ,X_{d}) $ in Assumptions 3.1 and 3.2 guarantee the existence of the TM and TCM. The dependence structure defined by (3.1) and (3.2) was first proposed in Mitra and Resnick [Reference Mitra and Resnick37] and has been extensively studied and applied in risk theory; see Asimit et al. [Reference Asimit, Furman, Tang and Vernic2], Hashorva and Li [Reference Hashorva and Li21], and Asimit and Li [Reference Asimit and Li3]. Since $h(t)=o(t)$ , the relation (3.1) obviously implies pairwise asymptotic independence among $X_{1}$ , …, $X_{d}$ .

For brevity, in what follows we write

(3.3) \begin{equation}C_{i}=\frac{c_{i}}{\sum_{j=1}^{d}c_{j}}\in \left( 0,1\right) ,\quad1\leq i\leq d, \end{equation}

where $c_{1}$ , …, $c_{d}$ are the positive constants from Assumption 3.1 or 3.2. Now we are ready to state our main results.

Theorem 3.1. Consider the TM defined by (1.1) with $n\in \textbf{N}^{+}$ .

(i) Under Assumption 3.1, it holds for each $1\leq k\leq d$ that

(3.4) \begin{equation}\textrm{TM}_{k}^{(n)}(t)\sim \frac{\alpha }{\alpha -n}C_{k}t^{n}.\end{equation}

(ii) Under Assumption 3.2, it holds for each $1\leq k\leq d$ that

(3.5) \begin{equation}\textrm{TM}_{k}^{(n)}(t)\sim C_{k}t^{n}. \end{equation}

Recall the TCM defined by (1.2). By the binomial expansion theorem, we have

\begin{equation*}\textrm{TCM}_{k}^{(n)}(t)=\sum_{i=0}^{n-1}\left(\begin{array}{c}n \\i\end{array}\right) ({-}1)^{i}\left( \textrm{TM}_{k}^{(1)}(t)\right) ^{i}\textrm{TM}_{k}^{(n-i)}(t)+({-}1)^{n}\left( \textrm{TM}_{k}^{(1)}(t)\right) ^{n},\end{equation*}

where $\left(\begin{array}{c}n \\i\end{array}\right) =n!/\left( i!(n-i)!\right) $ . Then, applying Theorem 3.1 immediately yields the following corollary for the TCM.

Corollary 3.1. Consider the TCM defined by (1.2) with $n\in \textbf{N}^{+}$ .

(i) Under Assumption 3.1, it holds for each $1\leq k\leq d$ that

(3.6) \begin{equation}\textrm{TCM}_{k}^{(n)}(t)=\left( A_{\alpha ,n,k}+o(1)\right) t^{n},\end{equation}

where

(3.7) \begin{equation}A_{\alpha ,n,k}=\sum_{i=0}^{n-1}\left(\begin{array}{c}n \\i\end{array}\right) ({-}1)^{i}\frac{\alpha ^{i+1}}{\left( \alpha -1\right)^{i}\left(\alpha -n+i\right) }C_{k}^{i+1}+({-}1)^{n}\left( \frac{\alpha }{\alpha -1}\right) ^{n}C_{k}^{n}. \end{equation}

(ii) Under Assumption 3.2, it holds for each $1\leq k\leq d$ that

(3.8) \begin{equation}\textrm{TCM}_{k}^{(n)}(t)=\left( A_{n,k}+o(1)\right) t^{n},\end{equation}

where

(3.9) \begin{equation}A_{n,k}=\lim_{\alpha \rightarrow \infty }A_{\alpha ,n,k}=C_{k}\left(1-C_{k}\right) \left( \left( 1-C_{k}\right)^{n-1}+({-}1)^{n}C_{k}^{n-1}\right) . \end{equation}

Remark 3.1. The relations (3.4) and (3.5) imply that $\textrm{TM}_{k}^{(n)}(t)\rightarrow \infty $ for any $n\in \textbf{N}^{+}$ under our models. Hence, when t is large enough, $\textrm{TM}_{k}^{(n)}(t)$ and $\textrm{TCM}_{k}^{(2n)}(t)$ define two standard measures with non-negativity. Moreover, it is easy to see from (3.9) that $A_{2n,k}>0$ , since $0<C_{k}<1$ . Actually, we can also prove that

(3.10) \begin{equation}A_{\alpha ,2n,k}>0 \end{equation}

under the conditions of Corollary 3.1(i); see Section 5. Therefore, (3.6) and (3.8) provide us with precise asymptotic estimates of $\textrm{TCM}_{k}^{(2n)}(t)$ . On the other hand, both $A_{\alpha ,2n-1,k}$ and $A_{2n-1,k}$ may be less than or equal to 0 for certain choices of $\alpha $ and $C_{k}$ . When $A_{\alpha ,2n-1,k}<0$ or $A_{2n-1,k}<0$ , (3.6) or (3.8) implies that $\textrm{TCM}_{k}^{(2n-1)}(t)$ tends to $-\infty $ and hence is not a non-negative measure for large t. In case $A_{\alpha ,2n-1,k}=0$ or $A_{2n-1,k}=0$ , (3.6) or (3.8) fails to give a precise asymptotic result for $\textrm{TCM}_{k}^{(2n-1)}(t)$ . Seeking precise estimates of $\textrm{TCM}_{k}^{(2n-1)}(t)$ in such a case requires more nuanced analysis, and we will not focus on it in the present paper.

Further consider the special case where, as mentioned before, the threshold t is chosen to be $\textrm{VaR}_{q}(S_{d})$ with $q\in (0,1)$ . By Theorem 3.1 of Chen and Yuen [Reference Chen and Yuen12] and Theorem 3.1 of Hashorva and Li [Reference Hashorva and Li21], we have

(3.11) \begin{equation}\mathbb{P}\!\left( S_{d}>t\right) \sim \mathbb{P}\!\left(\sum_{i=1}^{d}X_{i}^{+}>t\right) \sim\sum_{i=1}^{d}\overline{F}_{i}(t)\sim \left(\sum_{i=1}^{d}c_{i}\right) \overline{F}_{1}(t) \end{equation}

under Assumption 3.1 or 3.2. Then, using Lemma 2.1 of Asimit et al. [Reference Asimit, Furman, Tang and Vernic2] gives that

(3.12) \begin{equation}\textrm{VaR}_{q}(S_{d})\sim \left( \sum_{i=1}^{d}c_{i}\right) ^{1/\alpha }\textrm{VaR}_{q}(X_{1}),\quad q\uparrow 1, \end{equation}

under Assumption 3.1, while using Lemma 2.4 and the analysis under Corollary 3.2 of Asimit et al. [Reference Asimit, Furman, Tang and Vernic2] gives that

(3.13) \begin{equation}\textrm{VaR}_{q}(S_{d})\sim \textrm{VaR}_{1-(1-q)/\sum_{i=1}^{d}c_{i}}(X_{1}),\quad q\uparrow 1, \end{equation}

under Assumption 3.2. Hence, plugging (3.12) into (3.4) and (3.6) and plugging (3.13) into (3.5) and (3.8), we obtain the following asymptotic results for the corresponding TM and TCM as $q\uparrow 1$ .

Corollary 3.2. Consider the TM and TCM defined by (1.1) and (1.2) with $n\in \textbf{N}^{+}$ and $t=\textrm{VaR}_{q}(S_{d})$ .

(i) Under Assumption 3.1, we have, for each $1\leq k\leq d$ ,

\begin{equation*}\textrm{TM}_{k}^{(n)}\left( \textrm{VaR}_{q}(S_{d})\right) \sim\frac{\alpha }{\alpha -n}C_{k}\left( \sum_{i=1}^{d}c_{i}\right)^{n/\alpha }\left( \textrm{VaR}_{q}(X_{1})\right) ^{n},\quad q\uparrow 1,\end{equation*}

and

\begin{equation*}\textrm{TCM}_{k}^{(n)}\left( \textrm{VaR}_{q}(S_{d})\right) =\left(\left( \sum_{i=1}^{d}c_{i}\right) ^{n/\alpha }A_{\alpha,n,k}+o(1)\right) \left( \textrm{VaR}_{q}(X_{1})\right) ^{n},\quad q\uparrow 1,\end{equation*}

where $A_{\alpha ,n,k}$ is given by (3.7).

(ii) Under Assumption 3.2, we have, for each $1\leq k\leq d$ ,

\begin{equation*}\textrm{TM}_{k}^{(n)}\left( \textrm{VaR}_{q}(S_{d})\right) \sim C_{k}\left(\textrm{VaR}_{1-(1-q)/\sum\nolimits_{i=1}^{d}c_{i}}(X_{1})\right)^{n},\quad q\uparrow 1,\end{equation*}

and

\begin{equation*}\textrm{TCM}_{k}^{(n)}\left( \textrm{VaR}_{q}(S_{d})\right) =\left(A_{n,k}+o(1)\right) \left( \textrm{VaR}_{1-(1-q)/\sum\nolimits_{i=1}^{d}c_{i}}(X_{1})\right) ^{n},\quad q\uparrow 1,\end{equation*}

where $A_{n,k}$ is given by (3.9).

Remark 3.3. Furman and Landsman [Reference Furman and Landsman19] proposed the tail variance premium (TVP) and tail standard deviation premium (TSDP) principles for individual insurance risks. In terms of our notation, they are formulated as

\begin{equation*}\textrm{TVP}_{k}(t)=\textrm{TM}_{k}^{(1)}(t)+w\textrm{TCM}_{k}^{(2)}(t)\end{equation*}

and

\begin{equation*}\textrm{TSDP}_{k}(t)=\textrm{TM}_{k}^{(1)}(t)+w\sqrt{\textrm{TCM}_{k}^{(2)}(t)},\end{equation*}

where w is a non-negative constant; see Definition 3 of Furman and Landsman [Reference Furman and Landsman19]. Applying Theorem 3.1 and Corollary 3.1 immediately yields that

\begin{equation*}\textrm{TVP}_{k}(t)\sim w\textrm{TCM}_{k}^{(2)}(t)\sim w\!\left(\frac{\alpha}{\alpha -2}C_{k}-\frac{\alpha ^{2}}{\left( \alpha -1\right) ^{2}}C_{k}^{2}\right) t^{2}\end{equation*}

and

\begin{equation*}\textrm{TSDP}_{k}(t)\sim \left( \frac{\alpha }{\alpha -1}C_{k}+w\sqrt{\frac{\alpha }{\alpha -2}C_{k}-\frac{\alpha ^{2}}{\left( \alpha -1\right) ^{2}}C_{k}^{2}}\right) t\end{equation*}

under Assumption 3.1, and that

\begin{equation*}\textrm{TVP}_{k}(t)\sim w\textrm{TCM}_{k}^{(2)}(t)\sim wC_{k}\left(1-C_{k}\right) t^{2}\end{equation*}

and

\begin{equation*}\textrm{TSDP}_{k}(t)\sim \left( C_{k}+w\sqrt{C_{k}\left( 1-C_{k}\right) }\right) t\end{equation*}

under Assumption 3.2.

Remark 3.4. Our results can also be applied to find asymptotic solutions for some optimal capital allocation problems based on tail moment models. Denote by $p_{k}$ the capital allocated to $X_{k}$ for each $1\leq k\leq d$ , and by $p=\sum_{i=1}^{d}p_{i}$ the total capital, which is a fixed number. Dhaene et al. [Reference Dhaene, Tsanakas, Valdez and Vanduffel15] proposed a capital allocation criterion suggesting that the individual capital $p_{k}$ be set as close as possible to $X_{k}$ , in the sense of minimizing some distance measure. Xu and Mao [Reference Xu and Mao44] extended the idea of Dhaene et al. [Reference Dhaene, Tsanakas, Valdez and Vanduffel15] to a tail mean-variance model. Here we choose a quadratic distance measure and consider the following reduced version of the optimization problem studied in Xu and Mao [Reference Xu and Mao44]:

(3.14) \begin{equation}\min_{p_{1}(t),\ldots ,p_{d}(t)}\sum_{i=1}^{d}\mathbb{E}\!\left(\left. \left( X_{i}-p_{i}(t)\right) ^{2}\right\vert S_{d}>t\right),\quad \textrm{s.t.}\ \sum_{i=1}^{d}p_{i}(t)=p(t), \end{equation}

where $p_{1}(t)$ , …, $p_{d}(t)$ and p(t) are the individual capitals and total capital corresponding to the threshold t. By Theorem 2.2 of Xu and Mao [Reference Xu and Mao44], the optimal solution of (3.14) is

(3.15) \begin{equation}p_{k}^{\ast }(t)=\frac{p(t)-\sum_{i=1}^{d}\textrm{TM}_{i}^{(1)}(t)}{d}+\textrm{TM}_{k}^{(1)}(t),\quad 1\leq k\leq d. \end{equation}

Note that (3.15) holds for any $X_{1}$ , …, $X_{d}$ with finite second-order moments that make the problem (3.14) meaningful. Now, given the conditions of Theorem 3.1, the asymptotic estimate of $p_{k}^{\ast }(t)$ depends on the asymptotic behavior of p(t). Assume that $p(t)$ is set to be asymptotically proportionally equivalent to the conditional tail expectation of $S_{d}$ , i.e., for some $H\geq 1$ ,

\begin{equation*}p(t)\sim H\mathbb{E}\!\left( S_{d}\left\vert S_{d}>t\right. \right)=H\sum_{i=1}^{d}\textrm{TM}_{i}^{(1)}(t).\end{equation*}

Then, applying Theorem 3.1 to (3.15) gives that

\begin{equation*}p_{k}^{\ast }(t)\sim \frac{\alpha }{\alpha -1}\left( \frac{H-1}{d}+C_{k}\right) t\end{equation*}

under Assumption 3.1, and that

\begin{equation*}p_{k}^{\ast }(t)\sim \left( \frac{H-1}{d}+C_{k}\right) t\end{equation*}

under Assumption 3.2. When $H=1$ , so that $p(t)\sim\mathbb{E}\!\left(S_{d}\left\vert S_{d}>t\right. \right) $ , it holds under both Assumptions 3.1 and 3.2 that

\begin{equation*}p_{k}^{\ast }(t)\sim \textrm{TM}_{k}^{(1)}(t),\end{equation*}

which coincides with the Euler rule. Actually, in the case of $H=1$ , the asymptotic optimal solution $\left( p_{1}^{\ast }(t),\ldots,p_{d}^{\ast }(t)\right) $ minimizes each term in the summation of the optimization objective in (3.14). Moreover, if p(t) is set to be a quantity such that $t=o(p(t))$ , say, a linear combination of some higher-order TM and TCM, then it holds under both Assumptions 3.1 and 3.2 that

\begin{equation*}p_{1}^{\ast }(t)\sim \cdots \sim p_{d}^{\ast }(t)\sim\frac{1}{d}p(t).\end{equation*}

In this case, the asymptotic optimal allocation rule is just to allocate the total capital equally to each individual risk.

4. Numerical study on relative errors

In this section, we study the relative errors between our main asymptotic results and the accurate values corresponding to them when there are two individual risks (i.e., $d=2$ ) in the system. For this purpose, denote by $\widetilde{\textrm{TM}}_{k}^{(n)}(t)$ and $\widetilde{\textrm{TCM}}_{k}^{(n)}(t)$ with $k=1,2$ the asymptotic results obtained in Theorem 3.1 and Corollary 3.1 for $\textrm{TM}_{k}^{(n)}(t)$ and $\textrm{TCM}_{k}^{(n)}(t)$ . Denote by $R_{\textrm{TM},k}^{(n)}(t)$ and $R_{\textrm{TCM},k}^{(n)}(t)$ the corresponding relative errors, i.e.,

(4.1) \begin{equation}R_{\textrm{TM},k}^{(n)}(t)=\frac{\left\vert \textrm{TM}_{k}^{(n)}(t)-\widetilde{\textrm{TM}}_{k}^{(n)}(t)\right\vert}{\textrm{TM}_{k}^{(n)}(t)} \end{equation}

and

(4.2) \begin{equation}R_{\textrm{TCM},k}^{(n)}(t)=\frac{\left\vert \textrm{TCM}_{k}^{(n)}(t)-\widetilde{\textrm{TCM}}_{k}^{(n)}(t)\right\vert }{\textrm{TCM}_{k}^{(n)}(t)}. \end{equation}

For simplicity, let the random risks $X_{1}$ and $X_{2}$ be non-negative in this section. We assume that the joint distribution of $(X_{1},X_{2})$ belongs to the Farlie–Gumbel–Morgenstern (FGM) family with a parameter $\theta \in [{-}1,1]$ , i.e.,

(4.3) \begin{equation}\mathbb{P}\!\left( X_{1}\leq x,X_{2}\leq y\right)=F_{1}(x)F_{2}(y)\left( 1+\theta\overline{F}_{1}(x)\overline{F}_{2}(y)\right) . \end{equation}

It follows from (4.3) that

(4.4) \begin{equation}\mathbb{P}\!\left( X_{1}>x,X_{2}>y\right) =\overline{F}_{1}(x)\overline{F}_{2}(y)\left( 1+\theta F_{1}(x)F_{2}(y)\right) , \end{equation}

which implies asymptotic independence between $X_{1}$ and $X_{2}$ . Note that $X_{1}$ and $X_{2}$ are positively dependent if $\theta \geq 0$ , and are negatively dependent if $\theta \leq 0$ ; see Definition 5.2.1 of Nelsen [Reference Nelsen38].

In what follows, we give some numerical analyses in two specific scenarios to illustrate the effects of different model parameters on the convergence rates of $R_{\textrm{TM},1}^{(n)}(t)$ and $R_{\textrm{TCM},1}^{(2)}(t)$ .

Scenario 4.1. Let $(X_{1},X_{2})$ have the joint distribution given by (4.3) with the marginal distributions

\begin{equation*}F_{i}(x)=1-\lambda _{i}^{\alpha }\left( x+\lambda _{i}\right)^{-\alpha },\quad x\geq 0,\ \alpha >0,\ i=1,2,\end{equation*}

where $\lambda _{1}$ and $\lambda _{2}$ are set to be 100 and 120, respectively.

In this scenario, it is easy to check that Assumption 3.1 holds with $c_{1}=1$ and $c_{2}=\left( \lambda _{2}/\lambda _{1}\right) ^{\alpha}$ . For choices of $\alpha $ and n such that $\alpha >n$ , applying Theorem 3.1(i) gives that

(4.5) \begin{equation}\widetilde{\textrm{TM}}_{1}^{(n)}(t)=\frac{\alpha }{\left( \alpha-n\right) \left( 1+\left( \lambda _{2}/\lambda _{1}\right) ^{\alpha}\right) }t^{n}. \end{equation}

On the other hand, denote by $f_{i}$ the density of $X_{i}$ for $i=1,2$ , i.e.,

\begin{equation*}f_{i}(x)=\alpha \lambda _{i}^{\alpha }\left( x+\lambda _{i}\right)^{-\alpha -1},\quad x\geq 0.\end{equation*}

By (4.3), the joint density of $(X_{1},X_{2})$ is

\begin{equation*}f_{X_{1},X_{2}}(x,y)=f_{1}(x)f_{2}(y)\left( 1+\theta \!\left(1-2F_{1}(x)\right) \left( 1-2F_{2}(y)\right) \right) ,\quad x,y\geq0.\end{equation*}

Then, the joint density of $(X_{1},X_{1}+X_{2})$ can be calculated and has the form

\begin{equation*}f_{X_{1},X_{1}+X_{2}}(x,y)=f_{1}(x)f_{2}(y-x)\left( 1+\theta \!\left(1-2F_{1}(x)\right) \left( 1-2F_{2}(y-x)\right) \right) ,\quad 0\leq x\leq y.\end{equation*}

Further calculations yield that

\begin{align*}\mathbb{P}\!\left( X_{1}\in \textrm{d}x,X_{1}+X_{2}>t\right)& = \int_{t}^{\infty }f_{X_{1},X_{1}+X_{2}}(x,y)\textrm{d}y\textrm{d}x \nonumber\\& = f_{1}(x)\overline{F}_{2}(t-x)\left( 1-\theta \!\left(1-2F_{1}(x)\right) F_{2}(t-x)\right) \textrm{d}x,\end{align*}

and

\begin{align*}\mathbb{P}\!\left( X_{1}+X_{2}>t\right) & = \int_{0}^{\infty}\mathbb{P}\!\left(X_{1}\in \textrm{d}x,X_{1}+X_{2}>t\right) \\& = \int_{0}^{t}f_{1}(x)\overline{F}_{2}(t-x)\left( 1-\theta \!\left(1-2F_{1}(x)\right) F_{2}(t-x)\right)\textrm{d}x+\overline{F}_{1}(t).\end{align*}

Thus, we can obtain the exact value of $\textrm{TM}_{1}^{(n)}(t)$ using

(4.6) \begin{align}\textrm{TM}_{1}^{(n)}(t) & = \int_{0}^{\infty }x^{n}\mathbb{P}\!\left(\left.X_{1}\in \textrm{d}x\right\vert X_{1}+X_{2}>t\right) \notag \\& = \frac{\int_{0}^{t}x^{n}f_{1}(x)\overline{F}_{2}(t-x)\left(1-\theta \!\left( 1-2F_{1}(x)\right) F_{2}(t-x)\right)\textrm{d}x+\int_{t}^{\infty}x^{n}f_{1}(x)\textrm{d}x}{\int_{0}^{t}f_{1}(x)\overline{F}_{2}(t-x)\left(1-\theta \!\left( 1-2F_{1}(x)\right) F_{2}(t-x)\right) \textrm{d}x+\overline{F}_{1}(t)}. \end{align}

Then, plugging (4.5) and (4.6) into (4.1) gives $R_{\textrm{TM},1}^{(n)}(t)$ . We calculate the values of $R_{\textrm{TM},1}^{(n)}(t)$ from $t=1000$ to $t=3000$ in steps of 50 for different choices of $\alpha $ , $\theta$ , and n, respectively, with the other parameters fixed. The corresponding numerical results are plotted in Figure 1. It is worth noting that the parameters $\alpha $ , $\theta$ , and n describe the tail behavior of the risks, the degree of the dependence, and the order of the TM, respectively. Thus, Figure 1(i) indicates that a heavier tail of $X_{1}$ , i.e., a smaller value of $\alpha $ , implies a faster convergence rate for $R_{\textrm{TM},1}^{(1)}(t)$ . Additionally, Figure 1(ii) shows that the convergence rate of $R_{\textrm{TM},1}^{(1)}(t)$ increases as the value of $\theta $ increases. In other words, the positive dependence between the risks may help to speed up the convergence of $R_{\textrm{TM},1}^{(1)}(t)$ , while the case of negative dependence is just the opposite. On the other hand, however, Figure 1(iii) seems inadequate to reveal a clear change law for $R_{\textrm{TM},1}^{(n)}(t)$ with respect to the order parameter n.

Figure 1. The graph of $R_{\textrm{TM},1}^{(n)}(t)$ in Scenario 4.1 with different parameters.

For $R_{\textrm{TCM},1}^{(n)}(t)$ , we consider here only $R_{\textrm{TCM},1}^{(2)}(t)$ , which corresponds to the interesting TV risk measure. By Corollary 3.1(i), it holds for $\alpha >2$ that

(4.7) \begin{equation}\widetilde{\textrm{TCM}}_{1}^{(2)}(t)=\left( \frac{\alpha }{\left(\alpha -2\right) \left( 1+\left( \lambda _{2}/\lambda _{1}\right)^{\alpha }\right) }-\frac{\alpha ^{2}}{\left( \alpha -1\right)^{2}\left( 1+\left( \lambda _{2}/\lambda _{1}\right) ^{\alpha}\right) ^{2}}\right) t^{2}. \end{equation}

The exact value of $\textrm{TCM}_{1}^{(2)}(t)$ can be calculated by using (4.6) and the equality

(4.8) \begin{equation}\textrm{TCM}_{1}^{(2)}(t)=\textrm{TM}_{1}^{(2)}(t)-\left( \textrm{TM}_{1}^{(1)}(t)\right) ^{2}. \end{equation}

Then, $R_{\textrm{TCM},1}^{(2)}(t)$ can be obtained by plugging (4.7) and (4.8) into (4.2). We calculate the values of $R_{\textrm{TCM},1}^{(2)}(t)$ from $t=1000$ to $t=3000$ in steps of 50 for different choices of $\alpha $ and $\theta $ , respectively, and present the corresponding numerical results in Figure 2. In contrast to the phenomenon shown in Figure 1(i) for $R_{\textrm{TM},1}^{(1)}(t)$ , Figure 2(i) indicates that the lighter the tail of $X_{1}$ is, the faster $R_{\textrm{TCM},1}^{(2)}(t)$ converges to 0. On the other hand, we cannot make a clear judgment on the change law of $R_{\textrm{TCM},1}^{(2)}(t)$ with respect to the dependence parameter $\theta $ according to Figure 2(ii).

Figure 2. The graph of $R_{\textrm{TCM},1}^{(n)}(t)$ in Scenario 4.1 with different parameters.

Scenario 4.2. Let $(X_{1},X_{2})$ have the joint distribution given by (4.3) with the marginal distributions

\begin{equation*}F_{1}(x)=F_{2}(x)=1-\textrm{e}^{-\left( \log \left( x+1\right)\right) ^{\gamma }},\quad x\geq 0,\ \gamma >1.\end{equation*}

In this scenario, we have $F_{1}\in \textrm{GMDA}(h)$ with

\begin{equation*}h(t)\sim \frac{t}{\gamma \left( \log t\right) ^{\gamma -1}}.\end{equation*}

Since $h(t)\rightarrow \infty $ , (3.1) follows from (4.4). It is not difficult to verify that

\begin{equation*}\overline{F}_{1}^{2}\left( h\!\left( t\right) \right) =o\!\left( \overline{F}_{1}\left( t\right) \right),\end{equation*}

which, combined with (4.4), implies (3.2). Thus, Assumption 3.2 holds with $c_{1}=c_{2}=1$ . Applying Theorem 3.1(ii) gives that

(4.9) \begin{equation}\widetilde{\textrm{TM}}_{1}^{(n)}(t)=\frac{1}{2}t^{n}. \end{equation}

The exact value of $\textrm{TM}_{1}^{(n)}(t)$ can be obtained using (4.6) with

\begin{equation*}f_{1}(x)=\frac{\gamma \left( \log \left( x+1\right) \right) ^{\gamma -1}\textrm{e}^{-\left( \log \left( x+1\right) \right) ^{\gamma}}}{x+1},\quad x\geq 0.\end{equation*}

Then, plugging (4.9) and (4.6) into (4.1) gives $R_{\textrm{TM},1}^{(n)}(t)$ . Additionally, by Corollary 3.1(ii), we have

\begin{equation*}\widetilde{\textrm{TCM}}_{1}^{(2)}(t)=\frac{1}{4}t^{2},\end{equation*}

which, together with (4.8) and (4.2), gives $R_{\textrm{TCM},1}^{(2)}(t)$ . Similarly as in Scenario 4.1, we calculate the values of $R_{\textrm{TM},1}^{(n)}(t)$ and $R_{\textrm{TCM},1}^{(2)}(t)$ from $t=1000$ to $t=3000$ in steps of 50 for different choices of the model parameters; we present the corresponding numerical results in Figures 3 and 4. In this scenario, $X_{1}$ and $X_{2}$ have an identical rapidly varying tail, which is lighter than any regularly varying tails as considered in Scenario 4.1. The numerical results reflect different relationships between the model parameters and the convergence rates of $R_{\textrm{TM},1}^{(n)}(t)$ and $R_{\textrm{TCM},1}^{(2)}(t)$ . Figures 3(i, ii) and 4(i, ii) indicate that a lighter tail of $X_{1}$ or a larger value of $\theta $ may lead to faster convergence rates for both $R_{\textrm{TM},1}^{(1)}(t)$ and $R_{\textrm{TCM},1}^{(2)}(t)$ . Moreover, Figure 3(iii) shows that a smaller value of n tends to give a faster convergence rate for $R_{\textrm{TM},1}^{(n)}(t)$ .

Figure 3. The graph of $R_{\textrm{TM},1}^{(n)}(t)$ in Scenario 4.2 with different parameters.

Figure 4. The graph of $R_{\textrm{TCM},1}^{(n)}(t)$ in Scenario 4.2 with different parameters.

It should be clarified that all the observations from the numerical study above are based on the model setups and value range of t that we considered in Scenarios 4.1 and 4.2. Rigorous investigations of the convergence properties of $R_{\textrm{TM},k}^{(n)}(t)$ and $R_{\textrm{TCM},k}^{(n)}(t)$ would require us to seek analytical asymptotic expressions for $R_{\textrm{TM},k}^{(n)}(t)$ and $R_{\textrm{TCM},k}^{(n)}(t)$ with respect to the model parameters. This task is essentially related to second-order asymptotic results on $\textrm{TM}_{k}^{(n)}(t)$ and $\textrm{TCM}_{k}^{(n)}(t)$ , and we will not pursue it in this paper. The reader is referred to Section 5.1 of Li [Reference Li34] for analytical asymptotic results on $R_{\textrm{TM},1}^{(1)}(t)$ in some special cases of Scenario 4.1.

5. Proofs of the main results

We begin with two lemmas established under general frameworks, in which the distributions of the random variables are not assumed to be of any specific type.

Proof. Part (i): Clearly, for all $y\in [ \varepsilon ,1]$ ,

\begin{equation*}\mathbb{P}\!\left( Z_{k}>t,\sum_{i=1}^{d}Z_{i}>t\right) \leq\mathbb{P}\!\left( Z_{k}>yt,\sum_{i=1}^{d}Z_{i}>t\right) \leq\mathbb{P}\!\left( Z_{k}>\varepsilon t,\sum_{i=1}^{d}Z_{i}>t\right) .\end{equation*}

Thus, we only need to prove

(5.1) \begin{equation}\mathbb{P}\!\left( Z_{k}>\varepsilon t,\sum_{i=1}^{d}Z_{i}>t\right)\lesssim \overline{V}_{k}(t) \end{equation}

and

(5.2) \begin{equation}\mathbb{P}\left( Z_{k}>t,\sum_{i=1}^{d}Z_{i}>t\right) \gtrsim\overline{V}_{k}(t). \end{equation}

We have

(5.3) \begin{align}\mathbb{P}\!\left( Z_{k}>\varepsilon t,\sum_{i=1}^{d}Z_{i}>t\right) &\leq \mathbb{P}\!\left( Z_{k}>\varepsilon t,\sum_{i=1}^{d}Z_{i}^{+}>t\right) \notag\\& = \mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}^{+}>t\right)-\mathbb{P}\!\left(\sum_{i=1}^{d}Z_{i}^{+}>t,Z_{k}\leq \varepsilon t\right) \notag \\&=\!:\,\mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}^{+}>t\right)-I(t). \end{align}

It holds that

(5.4) \begin{align}I(t) &\geq \mathbb{P}\!\left( \bigcup_{i=1}^{d}\left\{Z_{i}>t\right\},Z_{k}\leq \varepsilon t\right) \notag \\& = \mathbb{P}\!\left( \bigcup_{i=1}^{d}\left\{ Z_{i}>t\right\} \right) -\mathbb{P}\!\left( \bigcup_{i=1}^{d}\left\{ Z_{i}>t\right\},Z_{k}>\varepsilon t\right) \notag \\&\geq \sum_{\substack{ i=1 \\ i\neq k}}^{d}\mathbb{P}\!\left(Z_{i}>t\right) -\sum_{1\leq i<j\leq d}\mathbb{P}\!\left(Z_{i}>t,Z_{j}>t\right) -\sum _{\substack{ i=1 \\ i\neq k}}^{d}\mathbb{P}\!\left( Z_{i}>t,Z_{k}>\varepsilon t\right),\end{align}

where in the last step we used the Bonferroni inequality and the fact that $\left\{ Z_{k}>t,Z_{k}>\varepsilon t\right\} =\left\{ Z_{k}>t\right\}$ , since $0<\varepsilon \leq 1$ . We then obtain (5.1) by plugging (5.4) into (5.3) and applying the conditions (a)–(c). On the other hand, we write

(5.5) \begin{align}\mathbb{P}\!\left( Z_{k}>t,\sum_{i=1}^{d}Z_{i}>t\right)& = \mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}>t\right) -\mathbb{P}\!\left(\sum_{i=1}^{d}Z_{i}>t,Z_{k}\leq t\right) \notag \\& =\!:\,\mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}>t\right) -J(t).\end{align}

It holds that

(5.6) \begin{align}J(t) & = \mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}>t,Z_{k}\leq t,\bigcup_{\substack{ i=1 \\ i\neq k}}^{d}\left\{ Z_{i}>t\right\} \right) +\mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}>t,\bigcap_{i=1}^{d}\left\{ Z_{i}\leq t\right\}\right) \notag \\&\leq \sum_{\substack{ i=1 \\ i\neq k}}^{d}\mathbb{P}\!\left(Z_{i}>t\right) +\mathbb{P}\!\left(\sum_{i=1}^{d}Z_{i}^{+}>t,\bigcap_{i=1}^{d}\left\{Z_{i}\leq t\right\} \right) \notag \\& =\sum_{\substack{ i=1 \\ i\neq k}}^{d}\mathbb{P}\!\left( Z_{i}>t\right) +\mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}^{+}>t\right) -\mathbb{P}\!\left(\bigcup_{i=1}^{d}\left\{ Z_{i}>t\right\} \right) \notag \\&\leq \mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}^{+}>t\right)-\mathbb{P}\!\left( Z_{k}>t\right) +\sum_{1\leq i<j\leq d}\mathbb{P}\!\left( Z_{i}>t,Z_{j}>t\right) . \end{align}

Plugging (5.6) into (5.5) and noting the conditions (a)–(c), we obtain (5.2) and complete the proof of the assertion (i).

Part (ii): We always have

\begin{equation*}\mathbb{P}\!\left( Z_{k}>s,\sum_{i=1}^{d}Z_{i}>t\right) \leq \overline{V}_{k}(s).\end{equation*}

By the assertion (i) with $\varepsilon =1$ , it holds uniformly for $s\in[t,\infty )$ that

\begin{equation*}\mathbb{P}\!\left( Z_{k}>s,\sum_{i=1}^{d}Z_{i}>t\right) \geq\mathbb{P}\!\left( Z_{k}>s,\sum_{i=1}^{d}Z_{i}>s\right) \sim\overline{V}_{k}(s).\end{equation*}

Combining the two estimates above, we obtain the assertion (ii).

Lemma 5.2. Let $Z_{1}$ , …, $Z_{d}$ be d real-valued random variables with distributions $V_{1}$ , …, $V_{d}$ . Assume that the condition (a) of Lemma 5.1 and the following conditions hold:

1. (b^′) For any $\epsilon >0$ and each pair $1\leq i\neq j\leq d$ ,

   \begin{equation*}\lim_{t\rightarrow \infty }\frac{\mathbb{P}\!\left( Z_{i}^{-}>\epsilon t,Z_{j}>t\right) +\mathbb{P}\!\left( Z_{i}>t,Z_{j}>t\right) }{\overline{V}_{1}(t)}=0.\end{equation*}

2. (c^′) $\mathbb{P}\!\left(\sum_{i=1}^{d}Z_{i}^{+}>t\right) \sim\sum_{i=1}^{d}\overline{V}_{i}(t)$ .

Then, for each $1\leq k\leq d$ and any $\varepsilon >0$ , it holds uniformly for $y\in [ \varepsilon ,\infty )$ that

\begin{equation*}\mathbb{P}\!\left( Z_{k}^{-}>yt,\sum_{i=1}^{d}Z_{i}>t\right) =o\!\left(\overline{V}_{k}(t)\right) .\end{equation*}

Proof. It holds for all $y\in [ \varepsilon ,\infty )$ that

\begin{equation*}\mathbb{P}\!\left( Z_{k}^{-}>yt,\sum_{i=1}^{d}Z_{i}>t\right) \leq \mathbb{P}\!\left( Z_{k}^{-}>\varepsilon t,\sum_{i=1}^{d}Z_{i}^{+}>t\right) .\end{equation*}

Then, using steps similar to those shown in (5.3) and (5.4) and noting the fact that $\left\{ Z_{k}>t,Z_{k}^{-}>\varepsilon t\right\}=\varnothing $ , we have

\begin{align*}\mathbb{P}\!\left( Z_{k}^{-}>\varepsilon t,\sum_{i=1}^{d}Z_{i}^{+}>t\right)&\leq \mathbb{P}\!\left( \sum_{i=1}^{d}Z_{i}^{+}>t\right) -\sum_{i=1}^{d}\mathbb{P}\!\left( Z_{i}>t\right) \\&+\sum_{1\leq i<j\leq d}\mathbb{P}\!\left( Z_{i}>t,Z_{j}>t\right)+\sum _{\substack{ i=1 \\ i\neq k}}^{d}\mathbb{P}\!\left(Z_{i}>t,Z_{k}^{-}>\varepsilon t\right) .\end{align*}

Applying the conditions (a), (b $^{\prime }$ ), and (c $^{\prime }$ ) to the above relation completes the proof.

The next two lemmas can be regarded as a decomposition of the proof of Theorem 3.1. Through these two lemmas, we also find that the left tails of the individual risks do not affect the asymptotic properties of the TM and TCM under our models.

Proof. By Theorem 3.1 of Chen and Yuen [Reference Chen and Yuen12] and Theorem 3.1 of Hashorva and Li [Reference Hashorva and Li21], the equivalence relations shown in (3.11) hold under the conditions of this lemma. Recalling also (2.7) and (3.1), it is easy to check that all the conditions of Lemma 5.1 are satisfied by $(X_{1},\ldots ,X_{d})$ , and hence the assertions obtained in Lemma 5.1 hold for $(X_{1},\ldots ,X_{d})$ . For any $0<\varepsilon <1$ , we write

(5.7) \begin{align}\textrm{TM}_{k,+}^{(n)}(t) & = \left( \int_{0}^{\varepsilon t^{n}}+\int_{\varepsilon t^{n}}^{t^{n}}+\int_{t^{n}}^{\infty}\right) \mathbb{P}\!\left( X_{k}>x^{1/n}\left\vert S_{d}>t\right.\right) \textrm{d}x\notag \\&=\!:\,I_{1}(t)+I_{2}(t)+I_{3}(t). \end{align}

Clearly,

(5.8) \begin{equation}0\leq I_{1}(t)\leq \varepsilon t^{n}. \end{equation}

For $I_{2}(t)$ , we first note that, by (3.11) with $\overline{F}_{i}(t)\sim c_{i}\overline{F}_{1}(t)$ for each $1\leq i\leq d$ ,

(5.9) \begin{equation}\frac{\overline{F}_{i}(t)}{\mathbb{P}\!\left( S_{d}>t\right)}\rightarrow C_{i},\quad 1\leq i\leq d, \end{equation}

where $C_{i}$ is given by (3.3). Then, under Assumption 3.1 or 3.2, it holds that

(5.10) \begin{align}I_{2}(t) & = \frac{\int_{\varepsilon t^{n}}^{t^{n}}\mathbb{P}\!\left(X_{k}>x^{1/n},S_{d}>t\right) \textrm{d}x}{\mathbb{P}\!\left(S_{d}>t\right) }\notag \\&= \frac{\int_{\varepsilon }^{1}\mathbb{P}\!\left(X_{k}>ty^{1/n},S_{d}>t\right) \textrm{d}y}{\mathbb{P}\!\left( S_{d}>t\right) }t^{n} \notag \\&\sim \left( 1-\varepsilon \right) \frac{\overline{F}_{k}(t)}{\mathbb{P}\!\left( S_{d}>t\right) }t^{n} \notag \\&\sim \left( 1-\varepsilon \right) C_{k}t^{n}, \end{align}

where in the second step we used a change of variables, in the third step we used Lemma 5.1(i), and in the last step we used (5.9). Finally, under Assumption 3.1 or 3.2, we have

\begin{align*}I_{3}(t) &= \frac{\int_{t^{n}}^{\infty }\mathbb{P}\!\left(X_{k}>x^{1/n},S_{d}>t\right) \textrm{d}x}{\mathbb{P}\!\left(S_{d}>t\right) }\\&\sim \frac{\int_{t^{n}}^{\infty }\overline{F}_{k}(x^{1/n})\textrm{d}x}{\mathbb{P}\!\left( S_{d}>t\right) } \\& = \frac{n\int_{t}^{\infty }y^{n-1}\overline{F}_{k}(y)\textrm{d}y}{t^{n}\overline{F}_{k}(t)}\frac{\overline{F}_{k}(t)}{\mathbb{P}\!\left(S_{d}>t\right) }t^{n},\end{align*}

where in the second step we used Lemma 5.1(ii) and in the last step we used a change of variables. If Assumption 3.1 holds, then $t^{n-1}\overline{F}_{k}(t)\in \mathcal{R}_{-\alpha +n-1}$ with $-\alpha+n-1<-1$ . Hence, it follows from (5.9) and (2.3) that

(5.11) \begin{equation}I_{3}(t)\sim \frac{n}{\alpha -n}C_{k}t^{n}. \end{equation}

If Assumption 3.2 holds, then using (5.9) and (2.4) gives that

(5.12) \begin{equation}I_{3}(t)=o\!\left( t^{n}\right) . \end{equation}

Plugging (5.8), (5.10), and (5.11) or (5.12) into (5.7) and letting $\varepsilon \rightarrow 0$ , we complete the proof.

Lemma 5.4. Let $\textrm{TM}_{k,-}^{(n)}(t)=\mathbb{E}\!\left(\left( X_{k}^{-}\right) ^{n}\left\vert S_{d}>t\right. \right) $ for each $1\leq k\leq d$ . Under Assumption 3.1 or 3.2, we have

\begin{equation*}\textrm{TM}_{k,-}^{(n)}(t)=o\!\left( t^{n}\right) .\end{equation*}

Proof. In view of (3.11), it is easy to see that Lemma 5.2 is applicable for $(X_{1},\ldots ,X_{d})$ under Assumption 3.1 or 3.2. For any $0<\varepsilon <1$ , we write

(5.13) \begin{align}\textrm{TM}_{k,-}^{(n)}(t) & = \left( \int_{0}^{\varepsilon t^{n}}+\int_{\varepsilon t^{n}}^{\varepsilon^{-1}t^{n}}+\int_{\varepsilon ^{-1}t^{n}}^{\infty }\right)\mathbb{P}\!\left( X_{k}^{-}>x^{1/n}\left\vert S_{d}>t\right. \right) \textrm{d}x \notag \\&=\!:\,I_{1}(t)+I_{2}(t)+I_{3}(t). \end{align}

Clearly, the relation (5.8) still holds for $I_{1}(t)$ . In addition, under Assumption 3.1 or 3.2,

(5.14) \begin{align}I_{2}(t) & = \frac{\int_{\varepsilon t^{n}}^{\varepsilon ^{-1}t^{n}}\mathbb{P}\!\left( X_{k}^{-}>x^{1/n},S_{d}>t\right)\textrm{d}x}{\mathbb{P}\!\left(S_{d}>t\right) } \notag \\&\leq \frac{\mathbb{P}\!\left( X_{k}^{-}>\varepsilon ^{1/n}t,S_{d}>t\right) }{\overline{F}_{k}(t)}\frac{\overline{F}_{k}(t)}{\mathbb{P}\!\left(S_{d}>t\right) }\left( \varepsilon ^{-1}-\varepsilon \right) t^{n} \notag \\& = o\!\left( t^{n}\right) , \end{align}

where in the last step we used Lemma 5.2 and (5.9). Moreover, since $F_{k}({-}t)=O(\overline{F}_{1}(t))$ under Assumption 3.1 or 3.2, there is some constant B such that $\mathbb{P}\!\left(X_{k}^{-}>t\right) \leq B\overline{F}_{1}(t)$ for t large enough. Hence, we have

\begin{align*}I_{3}(t) & = \frac{\int_{\varepsilon ^{-1}t^{n}}^{\infty}\mathbb{P}\!\left( X_{k}^{-}>x^{1/n},S_{d}>t\right)\textrm{d}x}{\mathbb{P}\!\left(S_{d}>t\right) } \\&\leq \frac{\int_{\varepsilon ^{-1}t^{n}}^{\infty }\mathbb{P}\!\left(X_{k}^{-}>x^{1/n}\right) \textrm{d}x}{\mathbb{P}\!\left( S_{d}>t\right) } \\&\lesssim \frac{B\int_{\varepsilon ^{-1}t^{n}}^{\infty }\overline{F}_{1}\!\left( x^{1/n}\right) \textrm{d}x}{\mathbb{P}\!\left( S_{d}>t\right) } \\& = \frac{Bn\int_{\varepsilon ^{-1/n}t}^{\infty }y^{n-1}\overline{F}_{1}(y)\textrm{d}y}{\varepsilon ^{-1}t^{n}\overline{F}_{1}\left(\varepsilon^{-1/n}t\right) }\frac{\overline{F}_{1}\left( \varepsilon ^{-1/n}t\right) }{\mathbb{P}\!\left( S_{d}>t\right) }\varepsilon ^{-1}t^{n},\end{align*}

where in the last step we used a change of variables. If Assumption 3.1 holds, then using (2.3), $F_{1}\in\mathcal{R}_{-\alpha }$ , and (5.9) yields that

(5.15) \begin{equation}I_{3}(t)\lesssim \frac{Bn}{\alpha -n}C_{1}\varepsilon ^{\alpha/n-1}t^{n}. \end{equation}

If Assumption 3.2 holds, then it follows from (2.4) and (5.9) that

(5.16) \begin{equation}I_{3}(t)\lesssim \frac{Bn\int_{\varepsilon ^{-1/n}t}^{\infty }y^{n-1}\overline{F}_{1}(y)\textrm{d}y}{\varepsilon^{-1}t^{n}\overline{F}_{1}\left(\varepsilon ^{-1/n}t\right) }\frac{\overline{F}_{1}\left( t\right) }{\mathbb{P}\!\left( S_{d}>t\right) }\varepsilon ^{-1}t^{n}=o\!\left( t^{n}\right). \end{equation}

Plugging (5.8), (5.14), and (5.15) or (5.16) into (5.13) and letting $\varepsilon\rightarrow 0$ , we complete the proof.

Proof of Theorem 3.1. It is clear that

\begin{align*}\textrm{TM}_{k}^{(n)}(t) & = \mathbb{E}\!\left( \left(X_{k}^{+}-X_{k}^{-}\right) ^{n}\left\vert S_{d}>t\right. \right) \\& = \mathbb{E}\!\left( \left( X_{k}^{+}\right) ^{n}+({-}1)^{n}\left(X_{k}^{-}\right) ^{n}\left\vert S_{d}>t\right. \right) \\& = \textrm{TM}_{k,+}^{(n)}(t)+({-}1)^{n}\textrm{TM}_{k,-}^{(n)}(t).\end{align*}

A combination of Lemmas 5.3 and 5.4 indicates that

\begin{equation*}\textrm{TM}_{k,-}^{(n)}(t)=o(1)\textrm{TM}_{k,+}^{(n)}(t)\end{equation*}

under Assumption 3.1 or 3.2. Hence, we have

\begin{equation*}\textrm{TM}_{k}^{(n)}(t)\sim \textrm{TM}_{k,+}^{(n)}(t).\end{equation*}

Then the assertions (i) and (ii) of Lemma 5.3 imply the assertions (i) and (ii) of Theorem 3.1, respectively.

Proof of (3.10). Noting that the random variable $\left. \left( X_{k}-TM_{k}^{(1)}(t)\right) ^{2n}\right\vert D$ is non-negative for any event D, we have

\begin{align*}t^{2n} &\leq \mathbb{E}\!\left( \left. \left(X_{k}-TM_{k}^{(1)}(t)\right)^{2n}\right\vert X_{k}>t+TM_{k}^{(1)}(t),S_{d}>t\right) \\& = \frac{\int_{0}^{\infty }\mathbb{P}\!\left( \left(X_{k}-TM_{k}^{(1)}(t)\right)^{2n}>x,X_{k}>t+TM_{k}^{(1)}(t),S_{d}>t\right)\textrm{d}x}{\mathbb{P}\!\left( X_{k}>t+TM_{k}^{(1)}(t),S_{d}>t\right) } \\& \leq \frac{\int_{0}^{\infty }\mathbb{P}\!\left( \left(X_{k}-TM_{k}^{(1)}(t)\right) ^{2n}>x,S_{d}>t\right) \textrm{d}x}{\mathbb{P}\!\left( S_{d}>t\right) }\frac{\mathbb{P}\!\left( S_{d}>t\right) }{\mathbb{P}\!\left( X_{k}>t+TM_{k}^{(1)}(t),S_{d}>t\right) } \\& = \textrm{TCM}_{k}^{(2n)}(t)\frac{\mathbb{P}\!\left( S_{d}>t\right) }{\mathbb{P}\!\left( X_{k}>t+TM_{k}^{(1)}(t),S_{d}>t\right) }.\end{align*}

Thus,

\begin{align*}\textrm{TCM}_{k}^{(2n)}(t) &\geq \frac{\mathbb{P}\!\left(X_{k}>t+TM_{k}^{(1)}(t),S_{d}>t\right) }{\mathbb{P}\!\left( S_{d}>t\right) }t^{2n} \\&\sim \frac{\overline{F}_{k}\left( t+TM_{k}^{(1)}(t)\right) }{\mathbb{P}\!\left( S_{d}>t\right) }t^{2n},\end{align*}

where the last step follows from Lemma 5.1(ii). By Theorem 3.1(i), it holds that

\begin{equation*}TM_{k}^{(1)}(t)\sim \frac{\alpha }{\alpha -1}C_{k}t\leq \frac{2\alpha }{\alpha -1}C_{k}t.\end{equation*}

Then,

\begin{align*}\textrm{TCM}_{k}^{(2n)}(t) &\gtrsim \frac{\overline{F}_{k}\left( \left( 1+\frac{2\alpha }{\alpha -1}C_{k}\right) t\right) }{\mathbb{P}\!\left(S_{d}>t\right) }t^{2n} \\&\sim C_{k}\left( 1+\frac{2\alpha }{\alpha -1}C_{k}\right)^{-\alpha }t^{2n},\end{align*}

where in the last step we used $F_{k}\in \mathcal{R}_{-\alpha }$ and (5.9). Comparing the above estimate with (3.6) gives that

\begin{equation*}A_{\alpha ,2n,k}\geq C_{k}\left( 1+\frac{2\alpha }{\alpha-1}C_{k}\right) ^{-\alpha }>0.\end{equation*}

This completes the proof of (3.10).