Asymptotics for Weighted Random Sums

Mariana Olvera-Cravioto

paper

Asymptotics for Weighted Random Sums

arXiv:1102.0301

Abstract

Let ${X_{i}}$ be a sequence of independent identically distributed random variables with an intermediate regularly varying (IR) right tail $F ˉ$ . Let $(N, C_{1}, ..., C_{N})$ be a nonnegative random vector independent of the ${X_{i}}$ with $N \in N \cup {\infty}$ . 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 $P (M_{N} > x) \sim P (S_{N} > x) \sim E [i = 1 \sum N F ˉ (x / C_{i})],$ 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 $P (M_{N} > x) \sim P (S_{N} > x) \sim E [i = 1 \sum N F ˉ (x / C_{i})] + P (Z_{N} > x / E [X_{1}]) .$ For completeness, when the distribution of $Z_{N}$ is IR and heavier than $F ˉ$ , 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.