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| A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables |
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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 |
| Page view： 745
Net amount： 543 |
Authors： |
Fengyang CHENG; |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11401415). |
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| 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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