Asymptotics for Weighted Random Sums

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.