2. Some observations on the stochastic dominance

Throughout the paper, we work with random variables that are almost surely nonnegative.

The main focus of the paper is on studying random variables X such that

(SD) \begin{equation} X \le_\textrm{st} \theta_1 X_1 + \dots + \theta_n X_n\ \mbox{for all $\bar{\theta} \in \Delta_n$,}\end{equation}

where $X_1, \dots, X_n$ are independent or negatively dependent with the marginal laws equal to X (see Section 3.2 for the precise definition of negative dependence). We will also say that a distribution F satisfies property (SD) if a random variable $X\sim F$ satisfies it. If some of $\theta_1,\dots,\theta_n$ are 0, we can simply reduce the dimension of our problem. Therefore, for most of our results, we will assume $\bar{\theta}\in\Delta_n^+$ .

Since (SD) holds if a constant is added to X, we will, without loss of generality, only consider random variables with essential infimum 0. We will also be interested in distributions, and random variables, for which property (SD) holds with a strict inequality. Let us start by formulating and providing some straightforward observations of (SD).

1. (i) $\mathbb{E}(X) = \infty$ or X is a constant.

2. (ii) A random variable $aX + b$ with $a \ge 0$ and $b \in \mathbb{R}$ satisfies (SD).

3. (iii) Random variables $\max\{X,c\}$ and $\max\{X,Y\}$ satisfy (SD), with $c \ge 0$ .

4. (iv) A random variable g(X) with a convex nondecreasing function g satisfies (SD). In addition, if X satisfies (SD) with a strict inequality, g is convex and strictly increasing, then g(X) also satisfies (SD) with a strict inequality.

Properties (ii)–(iv) above demonstrate that, even if one knows only several random variables satisfying (SD), it is possible to construct many more. Of special interest is property (iii), which does not require any specific distributional properties of X and Y apart from property (SD).

3. A class of heavy-tailed distributions and stochastic dominance

In this section, we introduce a new class of heavy-tailed distributions. We explore several properties of this class and demonstrate that it contains many well-known distributions with infinite mean. We then prove that all distributions in this class satisfy property (SD). Along with the results of Proposition 1, this shows that the class of distributions satisfying property (SD) is large.

3.1 A class of heavy-tailed distributions

As has already been noted, we can, without loss of generality, consider random variables whose essential infimum is zero. For a random variable $X\sim F$ with $\textrm{ess{-}inf}\ X=0$ , we have $F(x)\gt 0$ for all $x\gt 0$ .

Definition 2. Let F be a distribution function with $\textrm{ess{-}inf}\ F=0$ and let $h_F(x)=-\log F(1/x)$ for $x\in(0,\infty)$ . We say that F belongs to $\mathcal H$ , denoted by $F\in\mathcal H$ , if $h_F$ is subadditive. We write $F\in\mathcal H_s$ if $h_F$ is strictly subadditive. For $X\sim F$ , we also write $X\sim \mathcal H$ (resp. $X\sim \mathcal H_s$ ) if $F\in\mathcal H$ (resp. $F\sim \mathcal H_s$ ).

Remark 1. By properties of subadditive functions (e.g., Theorems 7.2.4 and 7.2.5 of Hille and Phillips, Reference Hille and Phillips1996), $F\in \mathcal{H}$ if $h_F(x)/x$ is decreasing or $h_F$ is concave.

In the case of continuous distribution F, $F\in \mathcal H$ holds if and only if the survival function of $1/X$ is log-superadditive where $X\sim F$ . We will see later that all distributions in $\mathcal{H}$ have infinite mean and because of that we say $\mathcal{H}$ is a class of heavy-tailed distributions. Note that the definition of heavy-tailed distributions varies in different contexts; see, for example, Remark 3. Below are some examples in class $\mathcal{H}$ .

Example 1. (Fréchet distribution). For $\alpha\gt 0$ , the Fréchet distribution, denoted by Fréchet $(\alpha)$ , is defined as

\begin{align*}F(x)=\exp({-}x^{-\alpha}), x\gt 0.\end{align*}

If $\alpha\le 1$ , F has infinite mean. It is easy to check that $F\in\mathcal{H}$ if $\alpha\le 1$ and $F\in\mathcal{H}_s$ if $\alpha\lt 1$ , since for any $x,y\gt 0$ ,

\begin{align*}\frac{h_F(x)+h_F(y)}{h_F(x+y)}=\left(\frac{x}{x+y}\right)^\alpha+\left(1-\frac{x}{x+y}\right)^\alpha\ge 1.\end{align*}

As $h_F$ is additive when $\alpha=1$ , Fréchet(1) distribution can be thought as a “boundary” of class $\mathcal{H}$ .

Example 2 (Pareto(1) distribution). For $\alpha\gt 0$ , the Pareto distribution, denoted by Pareto $(\alpha)$ , is defined as

\begin{align*}F(x)=1-\frac{1}{(x+1)^\alpha}, x\gt 0.\end{align*}

Pareto $(\alpha)$ distributions have infinite mean if $\alpha\le 1$ . Taking second derivative of $h_F$ when $\alpha=1$ , we have $h''_F(x)=-1/(x+1)^2.$ Hence, $h_F$ is concave and $\textrm{Pareto}(1)\in\mathcal{H}_s$ .

We can show that Pareto $(\alpha)$ with $\alpha\le 1$ , as well as many other infinite-mean distributions in Table 1 also belong to $\mathcal{H}$ either directly using the definition or using some closure properties of $\mathcal{H}$ in Propositions 2 and 3 provided below; see Appendix for detailed derivations of examples in Table 1 and the proofs of Propositions 2 and 3.

Table 1.

Examples of distributions in $\mathcal{H}$ .

Proposition 2. Let $X\sim F$ where $F\in \mathcal{H}$ . The following statements hold.

1. (i) If F is strictly increasing on $[0,\infty)$ and $\mathbb{P}(X\lt \infty)=1$ , then F is continuous on $[0,\infty)$ .

2. (ii) For $\beta\gt 0$ , $F^\beta\in \mathcal{H}$ .

3. (iii) If, in addition, a random variable $Y\sim G$ , where $G\in\mathcal{H}$ , is independent of X, then $\max\{X,Y\}\in \mathcal{H}$ . In terms of distribution functions, if $F,G\in\mathcal{H}$ , then $FG\in\mathcal{H}$ .

4. (iv) For a nondecreasing, convex, and nonconstant function $f\,:\,\mathbb{R}_+\rightarrow \mathbb{R}_+$ with $f(0)=0$ , $f(X)\in\mathcal{H}$ .

Proposition 3. Let $\bar{\theta} \in \Delta_n^+$ . If distribution functions $F_1,\dots,F_n\in\mathcal{H}$ and $F_1\le_\textrm{st }\dots\le_\textrm{st }F_n$ , then $\sum_{i=1}^n\theta_iF_i\in\mathcal{H}$ .

It is clear that the various transforms of distributions in Proposition 2 from our class generate many different distributions, showing that the class $\mathcal H$ is indeed rather large. Suppose that $F_1,\dots,F_n$ are Pareto distributions with possibly different tail parameters $0\lt \alpha_1,\dots,\alpha_n\le 1$ . As $F_1,\dots,F_n$ are comparable in stochastic order, by Proposition 3, mixtures of $F_1,\dots,F_n$ are in $\mathcal{H}$ .

3.3 Main result

Theorem 1. If a random variable $X\in \mathcal{H}$ and random variables $X_1,\dots,X_n$ are NLOD with marginal laws equal to X, then for $\bar{\theta} \in\Delta_n^+$ ,

(3.1) \begin{equation} X \le_\textrm{st} \theta_1 X_1 + \dots + \theta_n X_n.\end{equation}

If $X\in\mathcal{H}_s$ , then $X \lt _\textrm{st}\sum_{i=1}^n\theta_{i}X_{i}$ .

Proof. Let $X\sim F$ and $\bar \theta\in\Delta_n^+$ . We have, for all $x\gt 0$ ,

\begin{align*}\mathbb{P}\left(\sum_{i=1}^n\theta_{i}X_{i}\le x\right)&\le \mathbb{P}(\theta_1X_1\le x,\dots,\theta_nX_n\le x) \le \prod_{i=1}^n F\left(\frac{x}{\theta_i}\right)= \prod_{i=1}^n \exp\left(-h_F\left(\frac{\theta_i}{x}\right)\right)\\&= \exp\left(- \sum_{i=1}^nh_F\left(\frac{\theta_i}{x}\right)\right)\le \exp\left(- h_F\left(\sum_{i=1}^n\frac{\theta_i}{x}\right)\right)=\exp\left(- h_F\left(\frac{1}{x}\right)\right)=F(x).\end{align*}

The strictness statement is straightforward. The proof is complete.

An immediate consequence of Theorem 1 and Proposition 1 (i) is that all distributions in $\mathcal{H}$ have infinite mean.

Remark 2 (Value-at-Risk). One regulatory risk measure in insurance and finance is Value-at-Risk (VaR). For a random variable $X\sim F$ and $p\in (0,1)$ , VaR is defined as $\mathrm{VaR}_{p}(X)=F^{-1}(p)$ . For two random variables X and Y, it is well known that $X\le_\textrm{st} Y$ if and only if $\mathrm{VaR}_p(X)\le \mathrm{VaR}_p(Y)$ for all $p\in(0,1)$ . Note that VaR is comonotonic-additive; a risk measure $\rho$ is comonotonic-additive if $\rho(Y+Z)=\rho(Y)+\rho(Z)$ for comonotonic random variables Y and Z.Footnote 1 Then we have $\mathrm{VaR}_p(X)=\sum_{i=1}^n\mathrm{VaR}_p(\theta_iX_i)$ for identically distributed random variables $X_1,\dots,X_n$ . By Theorem 1, superadditivity of VaR holds for $\bar{\theta}\in\Delta_n^+$ and identically distributed risks $X_1,\dots,X_n\in \mathcal{H}$ that are NLOD: For all $p\in(0,1)$ , $ \mathrm{VaR}_p(\theta_1 X_{1})+\dots+\mathrm{VaR}_p(\theta_n X_{n})\le \mathrm{VaR}_p(\theta_{1}X_{1}+\dots+\theta_{n}X_{n}).$ More generally, the superadditivity property holds for any comonotonic-additive risk measure that is consistent with stochastic order.

Remark 3 (Heavy-tailed distributions). A distribution F is said to be heavy-tailed in the sense of Falk et al. (Reference Falk, Hüsler and Reiss2011) with tail parameter $\alpha\gt 0$ , if $\overline F(x)= L(x)x^{-\alpha}$ where L is a slowly varying function, that is, $L(tx)/L(x)\rightarrow 1$ as $x\rightarrow \infty$ for all $t\gt 0$ . For iid heavy-tailed random variables $X_1,X_2,\dots\sim F$ , if there exist sequences of constants $\{a_n\}$ and $\{b_n\}$ where $b_n\gt 0$ such that $(\max\{X_1,\dots,X_n\}-a_n)/b_n$ converges to the Fréchet distribution, F is said to be in the maximum domain attraction of the Fréchet distribution. It is known in the Extreme Value Theory (Embrechts et al., Reference Embrechts, Klüppelberg and Mikosch1997) that a distribution is in the maximum domain of attraction of the Fréchet distribution if and only if the distribution is heavy-tailed. Note that for a heavy-tailed random variable X with $\alpha\le 1$ , $\mathbb{E}(|X|)=\infty$ . An interesting property of heavy-tailed risks with infinite mean is the asymptotic superadditivity of VaR: If $X_1,\dots,X_n$ are iid and heavy-tailed with tail parameter $\alpha\lt 1$ ,

\begin{align*}\lim_{p\rightarrow 1}\frac{\mathrm{VaR}_p(X_1+\dots+X_n)}{\mathrm{VaR}_p(X_1)+\dots+\mathrm{VaR}_p(X_n)}\gt 1.\end{align*}

See, for example, Example 3.1 of Embrechts et al. (Reference Embrechts, Lambrigger and Wüthrich2009) for the claim above. Heavy-tailed risks with infinite mean are not necessarily in $\mathcal{H}$ as the condition of $\mathcal{H}$ applies over the whole range of distributions, whereas heavy-tailed distributions have power-law shapes only in their tail parts. On the other hand, risks in $\mathcal{H}$ are not necessarily heavy-tailed in the sense of Falk et al. (Reference Falk, Hüsler and Reiss2011). For instance, the survival distributions of log-Pareto risks are slowly varying functions. Distributions with slowly varying tails are called super heavy-tailed.

Remark 4 (Convex order). Besides stochastic order, another popular notion of stochastic dominance to compare risks is convex order. For two random variables X and Y, X is said to be smaller than Y in convex order, denoted by $X\le_\textrm{cx}Y$ , if $\mathbb{E}(u(X))\le \mathbb{E}(u(Y))$ for all convex functions u provided that the expectations exist. The interpretation of $X\le_\textrm{cx}Y$ is that Y is more “spread-out” than X. If $X_1,\dots,X_n$ are iid and have a finite mean, by Theorem 3.A.35 of Shaked and Shanthikumar (Reference Shaked and Shanthikumar2007), for $\bar{\theta}\in\Delta_n^+$ , $\sum_{i=1}^n\theta_iX_i\le_\textrm{cx}X_1$ . Unlike Theorem 1, this leads to a diversification benefit. Note that $\le_\textrm{cx}$ is not suitable for the analysis of risks with infinite mean as the expectation of any increasing convex transform of these risks is infinity.

Remark 5 (Positive dependence). One may expect positive dependence to make larger values of the sum in (3.1) more likely and thus the sum more likely to stochastically dominate a single random variable. We believe that this intuition does not hold due to the very heavy tails of the random variables under consideration. It is known, for instance, that very large values of the sum of iid random variables with heavy tails are likely caused by a single random variable taking a large value, while other random variables are moderate. If random variables are positively dependent and some of them do not take large values, it makes others more likely to take moderate values too, hence positive dependence may hinder large values; see Alink et al. (Reference Alink, Löwe and Wüthrich2004) and Mainik and Rüschendorf (Reference Mainik and Rüschendorf2010) for such observations in some asymptotic senses. These phenomena can also be seen from the deadly risks considered by Müller (2024): For all $i\in[n]$ , $\mathbb{P}(X_i=0)=1-p$ and $\mathbb{P}(X_i=\infty)=p$ where $p\gt 0$ . For $\overline \theta\in\Delta_n^+$ , it is clear that $\sum_{i=1}^n\theta_iX_i=\infty$ as long as one of $X_1,\dots,X_n$ is $\infty$ . If $X_1,\dots,X_n$ are positively lower orthant dependent (PLOD), that is $\mathbb{P}(X_1\le x_1,\dots,X_n\le x_n)\ge \prod_{i=1}^n\mathbb{P}(X_i\le x_i)$ for all $x_1,\dots,x_n \in\mathbb{R}$ , then

\begin{align*}\mathbb{P}\left(\sum_{i=1}^n\theta_iX_i=\infty\right)&=1-\mathbb{P}\left(X_1=\dots=X_n=0\right),\\&=1-\mathbb{P}\left(X_1\le 0,\dots,X_n\le0\right)\le 1-\prod_{i=1}^n\mathbb{P}(X_i= 0).\end{align*}

Hence, $\sum_{i=1}^n\theta_iX_i$ is stochastically smaller when $X_1,\dots,X_n$ are PLOD compared to the case when $X_1,\dots,X_n$ are independent. The situation is reversed for NLOD random variables. However, (SD) still holds for PLOD risks $X_1,\dots,X_n$ as

\begin{align*}\mathbb{P}\left(\sum_{i=1}^n\theta_iX_i=\infty\right)=1-\mathbb{P}\left(X_1=\dots=X_n=0\right)\ge 1-\mathbb{P}(X_1=0)=\mathbb{P}(X_1=\infty).\end{align*}

In Chen et al. (Reference Chen, Hu, Wang and Zou2025b), (SD) is shown to hold for infinite-mean Pareto random variables that are positively dependent via some specific Clayton copula.

4. Weighted sums of non-identically distributed risks

In the previous section, property (SD) is studied for risks with the same marginal distribution. We now look at the case when risks are not necessarily identically distributed. Given non-identically distributed random variables $X_1, \dots, X_n$ and any $\bar{\theta}\in\Delta_n$ , the question is to study for which random variable X the following property holds

(4.1) \begin{equation} X \le_\textrm{st} \theta_1 X_1 + \dots + \theta_n X_n.\end{equation}

To study this problem, we introduce the class of super-Fréchet distributions defined below.

As convex transforms make the tail of random variables heavier, super-Fréchet distributions are more heavy-tailed than Fréchet(1) distribution, and thus the name. It is easy to check that a random variable X with $\textrm{ess-inf}\ X=0$ is super-Fréchet if and only if the function $g\,:\,x\mapsto 1/({-}\log \mathbb{P}(X\le x)) $ is strictly increasing and concave on $(0,\infty)$ with $\lim_{x\downarrow 0}g(x)=0$ .

As Fréchet(1) distribution is in $\mathcal{H}$ , by Proposition 1 (iv), Super-Fréchet distributions are in $\mathcal{H}$ . On the other hand, not all distributions in $\mathcal{H}$ are Super-Fréchet, which can be seen from the following example.

Example 3. For $c\gt 0$ , define a distribution function on $(0,\infty]$ :

\begin{align*}F(x)=\exp({-}c\lceil1/x\rceil),\mbox{for}\ x\in(0,\infty),\end{align*}

and $F(\infty)=1$ . Then $h_F(x)=c \lceil x\rceil $ , $x\in(0,\infty)$ , is subadditive, and thus $F\in \mathcal{H}$ . However, since F is not a continuous distribution, it is not super-Fréchet. The distributions in this example are the so-called inverse-geometric distributions, also considered in Example 2.7 of Arab et al. (Reference Arab, Lando and Oliveira2024).

Fréchet distributions with infinite mean, as well as many other distributions in the following example, are super-Fréchet.

Example 4. Pareto, Burr, paralogistic, and log-logistic random variables, all with infinite mean, are super-Fréchet distributions. Since all these random variables can be obtained by applying strictly increasing and convex transforms to Pareto(1) random variables (see Appendix C), it suffices to show that a Pareto(1) random variable is super-Fréchet. Write the Pareto(1) distribution as $F(x)=1-1/(x+1)=\exp({-}1/g(x))$ , $x\gt 0$ , where $g(x)=1/\log(1+1/x)$ . It is clear that g is strictly increasing and $\lim_{x\downarrow 0}g(x)=0$ . We show g is concave on $(0,\infty)$ . We have

\begin{align*}g''(x)=\frac{2 - (1 + 2 x) \log(1 + 1/x)}{x^2 (1 + x)^2 \log^3(1 + 1/x)}.\end{align*}

Let $r(x)=\log\left(1/x+1\right)-2/(1+2x)$ , $x\gt 0$ . It is easy to verify that r is strictly decreasing on $(0,\infty)$ and r(x) goes to 0 as x goes to infinity. Thus, $r(x)\gt 0$ and $g''(x)\lt 0$ for $x\in(0,\infty)$ .

We will assume $X_1,\dots,X_n$ in (4.1) are super-Fréchet. Since $X_1,\dots,X_n$ may not have the same distribution, how to choose the distribution of X is not clear. A perhaps natural candidate is the generalized mean of the distributions of $X_1,\dots,X_n$ . For $r\in \mathbb{R}\setminus \{0\}$ , $n\in \mathbb{N}$ , and $\textbf w=(w_1,\dots,w_n)\in \Delta_n$ , the generalized r-mean function is defined as

\begin{align*}M^{\textbf w}_r(u_1,\dots,u_n) = \left(w_1 u_1^r+\dots+w_n u_n^r \right)^{1/r}, (u_1,\dots,u_n) \in (0,\infty)^{n}.\end{align*}

The generalized 0-mean function is the weighted geometric mean, that is, $M^{\textbf w}_0(u_1,\dots,u_n) =\prod_{i=1}^n u_i^{w_i},$ which is also the limit of $M^{\textbf w}_r$ as $r\to 0$ . A generalized mean of distribution functions is a distribution function. In particular, if $r=1$ , it leads to a distribution mixture model, that is, if $X\sim M^{ \textbf w}_1(F_1,\dots,F_n)$ , X has the same distribution as $\sum_{i=1}^nX_i{\unicode{x1D7D9}}_{A_i}$ where $A_1,\dots,A_n$ are mutually exclusive, independent of $X_1,\dots,X_n$ , and $\mathbb{P}(A_i)=w_i$ for all $i\in[n]$ .

Theorem 2. If $X_1,\dots,X_n$ are super-Fréchet and NLOD with $X_i\sim F_i$ , $i\in[n]$ , and $X\sim M^{\bar \theta}_r(F_1,\dots,F_n)$ for some $r\ge 0$ , then for $\bar{\theta}\in\Delta_n^+$ ,

\begin{equation*}X \le_\textrm{st} \theta_1 X_1 + \dots + \theta_n X_n.\end{equation*}

Proof. Let $g_i(x)=1/({-}\log F_i(x))$ , $x\gt 0$ , for all $i\in[n]$ . As $g_i$ , $i\in[n]$ , is strictly increasing and concave on $(0,\infty)$ with $\lim_{x\downarrow 0}g_i(x)=0$ , $ g_i(x)\ge \theta g_i(x/\theta)$ for all $x\gt 0$ and $\theta\in(0,1)$ . Then, for $\theta\in(0,1)$ ,

(4.2) \begin{align} F_i\left(\frac{x}{\theta}\right)=\exp\left(-g_i\left(\frac{x}{\theta}\right)^{-1}\right)\le \exp\left(-\theta g_i(x)^{-1}\right)=F_i(x)^\theta. \end{align}

As $X_1,\dots,X_n$ are NLOD, by 4.2, for any $x\gt 0$ , $(\theta_1,\dots,\theta_n)\in\Delta_n$ , and $r\ge 0$ ,

\begin{align*}\mathbb{P}\left(\sum_{i=1}^n\theta_{i}X_{i}\le x\right)&\le \mathbb{P}(\theta_1X_1\le x,\dots,\theta_nX_n\le x)\le \prod_{i=1}^n F_i\left(\frac{x}{\theta_i}\right) \le\prod_{i=1}^n F_i\left(x\right)^{\theta_i}\\&=M^{\bar \theta}_0(F_1(x),\dots,F_n(x))\le M^{\bar \theta}_r(F_1(x),\dots,F_n(x))=\mathbb{P}(X\le x).\end{align*}

The last inequality is because the generalized mean function is monotone in r; that is, given any $\textbf w\in \Delta_n$ , $M^{\textbf w}_r\le M^{\textbf w}_s$ for $r\le s $ (Theorem 16 of Hardy et al., Reference Hardy, Littlewood and Pólya1934).

5. Comparison with existing results

In this section, we compare our results with the literature. We first consider the case when $X_1,\dots,X_n$ in (SD) are iid. In Arab et al. (Reference Arab, Lando and Oliveira2024), it is shown that (SD) holds for nonnegative random variables that are InvSub; a random variable $X\sim F$ and its distribution is called InvSub if $1-F(1/x)$ is subadditive. The class of InvSub distributions is larger than $\mathcal{H}$ as $h_F=-\log F(1/x)$ is subadditive implies that $1-F(1/x)$ is subadditive. Müller (2024) showed that (SD) holds for super-Cauchy random variables; a random variable $X\sim F$ and its distribution is called super-Cauchy if $F^{-1}(G(x))$ is convex where G is the standard Cauchy distribution function. Super-Cauchy distributions are continuous and can take positive values on the entire real line but they do not contain $\mathcal{H}$ as $\mathcal{H}$ includes non-continuous distributions (see Example 3). The proofs in both Arab et al. (Reference Arab, Lando and Oliveira2024) and Müller (2024) are short and elegant.

As our results cover the case of negatively dependent risks, for the rest of this section, we will focus on the comparison of our results with Chen et al. (Reference Chen, Embrechts and Wang2025a); to our best knowledge, Chen et al. (Reference Chen, Embrechts and Wang2025a) is the only other paper that deals with negatively dependent risks. Chen et al. (Reference Chen, Embrechts and Wang2025a) showed that

(5.1) \begin{equation} \textrm{(SD)}\ \mbox{holds for super-Pareto risks $X_1,\dots,X_n$ that are weakly negatively associated.} \end{equation}

For ease of comparison, definitions of super-Pareto distributions and weak negative association are given in a slightly different form from Chen et al. (Reference Chen, Embrechts and Wang2025a) below.

Definition 6. A set $S\subseteq \mathbb{R}^{k}$ , $k\in \mathbb{N}$ is decreasing if $\textbf x\in S$ implies $\textbf y\in S$ for all $\textbf y\le \textbf x$ . Random variables $X_1,\dots,X_n$ are weakly negatively associated if for any $i\in[n]$ , decreasing set $S \subseteq \mathbb{R}^{n-1}$ , and $x\in \mathbb{R}$ with $\mathbb{P}(X_i\le x)\gt 0$ ,

\begin{align*}\mathbb{P}(\textbf X_{-i} \in S, X_i\le x) \le \mathbb{P}(\textbf X_{-i}\in S)\mathbb{P}( X_i\le x).\end{align*}

where $\textbf X_{-i}=(X_1,\dots,X_{i-1}, X_{i+1},\dots,X_n)$ .

Proof. As Pareto(1) risks are in $\mathcal{H}$ (see Example 2), by Proposition 2 (iv), super-Pareto risks are in $\mathcal{H}$ . Since $X_1,\dots,X_n$ are weakly negatively associated, for any $(x_1,\dots,x_n)\in \mathbb{R}^n$ ,

\begin{align*}\mathbb{P}(X_1\le x_1,\dots,X_n\le x_n)\le \mathbb{P}(X_1\le x_1,\dots,X_{n-1}\le x_{n-1})\mathbb{P}(X_n\le n)\le \prod_{i=1}^n\mathbb{P}(X_i\le x_i).\end{align*}

Thus, $X_1,\dots,X_n$ are NLOD.

The above lemma shows that Theorem 1 implies (5.1), which is in Theorem 1 (i) of Chen et al.(Reference Chen, Embrechts and Wang2025a). We present below a corollary, which leads to a similar result as Theorem 1 (ii) of Chen et al. (Reference Chen, Embrechts and Wang2025a).

Corollary 1. Suppose that a random variable $X\in\mathcal{H}$ , random variables $X_1,\dots,X_n$ are NLOD with marginal laws equal to X, and $\xi_1,\dots,\xi_n$ are any positive random variables independent of $X,X_1,\dots,X_n$ with $\sum_{i=1}^n\xi_i\le 1$ . If $\mathbb{P}(cX\gt t) \ge c\mathbb{P}(X\gt t) $ for all $c \in (0,1]$ and $t\gt 0$ , then for $x\ge 0$ ,

(5.2) \begin{equation} \mathbb{P}\left(\sum_{i=1}^n\xi_i X_i \gt x\right) \ge \mathbb{E}\left(\sum_{i=1}^n \xi_i\right) \mathbb{P}(X\gt x).\end{equation}

Proof. By Theorem 1 and the independence between $\xi_1,\dots,\xi_n$ and $X,X_1,\dots,X_n$ , we have

\begin{align*}\mathbb{P}\left(\sum_{i=1}^n\xi_i X_i \gt x\right) &= \mathbb{E}\left[\mathbb{P}\left(\sum_{i=1}^n\xi_i X_i \gt x|(\xi_1,\dots,\xi_n)\right)\right]\\&\ge \mathbb{E}\left[\mathbb{P}\left(\left(\sum_{i=1}^n \xi_i\right) X \gt x|(\xi_1,\dots,\xi_n)\right)\right] \ge \mathbb{E}\left(\sum_{i=1}^n \xi_i\right) \mathbb{P}\left(X \gt x\right). \end{align*}

For $\bar \theta\in\Delta_n$ , let $A_1,\dots,A_n$ be any events independent of $(X_1,\dots,X_n)$ and event A be independent of X satisfying $\mathbb{P}(A)=\sum_{i=1}^n \theta_i\mathbb{P}(A_i)$ . If $X_1,\dots,X_n$ are financial losses, $A_1,\dots,A_n$ can be interpreted as the triggering events for these losses. Let $\xi_i = \theta_i {\unicode{x1D7D9}}_{A_i}$ . By (5.2), for $x\ge 0$ ,

\begin{align*}\mathbb{P}\left(\sum_{i=1}^n\theta_i X_i{\unicode{x1D7D9}}_{A_i} \gt x\right) \ge \mathbb{E}\left(\sum_{i=1}^n \theta_i {\unicode{x1D7D9}}_{A_i}\right) \mathbb{P}(X\gt x)=\mathbb{P}(A)\mathbb{P}(X\gt x)=\mathbb{P}(X{\unicode{x1D7D9}}_A\gt x),\end{align*}

which is equivalent to

(5.3) \begin{equation}X{\unicode{x1D7D9}}_A\le_\textrm{st}\sum_{i=1}^n\theta_i X_i{\unicode{x1D7D9}}_{A_i}.\end{equation}

Theorem 1 (ii) of Chen et al. (Reference Chen, Embrechts and Wang2025a) showed (5.3) with different assumptions from Corollary 1; we refer readers to Chen et al. (Reference Chen, Embrechts and Wang2025a) for more details.