A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

Abstract

The author obtains that the asymptotic relations

$$\mathbb{P}\left( {\sum\limits_{i = 1}^n {{\theta _i}{X_i}} >x} \right) \sim \mathbb{P}\left( {\mathop {\max }\limits_{1 \le m \le n} \sum\limits_{i = 1}^m {{\theta _i}{X_i}} >x} \right) \sim \mathbb{P}\left( {\mathop {\max {\theta _i}{X_i}}\limits_{1 \le i \le n} >x} \right) \sim \sum\limits_{i = 1}^n \mathbb{P}{\left( {{\theta _i}{X_i} >x} \right)}$$

hold as x → ∞, where the random weights θ₁,..., θ_n are bounded away both from 0 and from ∞ with no dependency assumptions, independent of the primary random variables X₁,..., X_n which have a certain kind of dependence structure and follow non-identically subexponential distributions. In particular, the asymptotic relations remain true when X₁,..., X_n jointly follow a pairwise Sarmanov distribution.