A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

Citation:

Fengyang CHENG.A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables[J].Chinese Annals of Mathematics B,2020,41(3):441~450
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Authors:

Fengyang CHENG;

Foundation:

This work was supported by the National Natural Science Foundation of China (No.11401415).
Abstract: The author obtains that the asymptotic relations \begin{align*} \mathbb{P}\Big(\sum_{i=1}^n \theta_iX_i >x\Big)\sim \mathbb{P}\Big(\max_{1\leq m\leq n}\sum_{i=1}^m \theta_i X_i>x\Big)\sim \mathbb{P}\Big(\max_{1\leq i\leq n}\theta_iX_i>x\Big)\sim \sum_{i=1}^n {\mathbb{P}( \theta_iX_i>x)} \end{align*} hold as $x\to\infty$, where the random weights $\theta_1,\cdots,\theta_n$ are bounded away both from $0$ and from $\infty$ with no dependency assumptions, independent of the primary random variables $X_1,\cdots,X_n$ which have a certain kind of dependence structure and follow non-identically subexponential distributions. In particular, the asymptotic relations remain true when $X_1,\cdots, X_n$ jointly follow a pairwise Sarmanov distribution.

Keywords:

Randomly weighted sums, Subexponential distributions, Ruin & probabilities, Insurance and financial risks

Classification:

60F15, 62P05
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