Randomly weighted sums of dependent subexponential random variables^*

Abstract

We consider the randomly weighted sums $\sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1}$, where $\left\{ {{X_k},1 \leqslant k \leqslant n} \right\}$ are n real-valued random variables with subexponential distributions, and $\left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\}$ are other n random variables independent of $\left\{ {{X_k},1 \leqslant k \leqslant n} \right\}$ and satisfying $a \leqslant \theta \leqslant b$ for some $0 < a \leqslant b < \infty$ and all $1 \leqslant k \leqslant n$. For $\left\{ {{X_k},1 \leqslant k \leqslant n} \right\}$ satisfying some dependent structures, we prove that

$$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$

as x → ∞.