Asymptotics for Weighted Random Sums

TL;DR

This paper analyzes the asymptotic behavior of weighted sums and their maxima involving IR-tailed random variables, providing conditions for their tail probabilities to be asymptotically equivalent.

Contribution

It establishes new asymptotic equivalences for the tail probabilities of weighted sums and maxima with IR tails, including cases with additional heavy-tailed sums.

Findings

Derived conditions for asymptotic equivalence of tail probabilities

Extended results to sums with positive mean and IR tail distributions

Identified when maxima and sums share similar tail behavior

Abstract

Let $\{X_i\}$ be a sequence of independent identically distributed random variables with an intermediate regularly varying (IR) right tail $\bar{F}$. Let $(N, C_1, ..., C_N)$ be a nonnegative random vector independent of the $\{X_i\}$ with $N \in \mathbb{N} \cup \{\infty\}$. We study the weighted random sum $S_N = \sum_{i=1}^N C_i X_i$, and its maximum, $M_N = \sup_{1 \leq k < N+1} \sum_{i=1}^k C_i X_i$. These type of sums appear in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which $$P(M_N > x) \sim P(S_N > x) \sim E[\sum_{i=1}^N \bar{F}(x/C_i)],$$ as $x \to \infty$. When $E[X_1] > 0$ and the distribution of $Z_N = \sum_{i=1}^N C_i$ is also IR, we obtain the asymptotics $$P(M_N > x) \sim P(S_N > x) \sim E[\sum_{i=1}^N \bar{F}(x/C_i)] + P(Z_N > x/E[X_1]).$$ For completeness, when the distribution of $Z_N$ is IR and heavier than $\bar{F}$, we also obtain conditions under which the asymptotic relations $$P(M_N > x) \sim P(S_N > x) \sim P(Z_N > x/E[X_1])$$ hold.