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 θ ₁,..., θ _n are bounded away both from 0 and from ∞ with no dependency assumptions, independent of the primary random variables X ₁,..., 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,..., X _n jointly follow a pairwise Sarmanov distribution.

Keywords

Randomly weighted sums / Subexponential distributions / Ruin probabilities / Insurance and financial risks

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