Asymptotics for Weighted Random Sums

Let $\{X_i\}$ be a sequence of independent identically distributed random variables with an intermediate regularly varying (IR) right tail $\bar{F}$. Let $(N,C_1,\mathrm{.}\mathrm{.}\mathrm{.},C_N)$ be a nonnegative random vector independent of the $\{X_i\}$ with $N\in \mathbb{N}\cup \{\mathrm{\infty }\}$. 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 $$P(M_N>x)\sim P(S_N>x)\sim E[\sum _{i=1}^N\bar{F}(x\mathrm{/}{C}_i)],$$ as $x\to \mathrm{\infty }$. When $E[X_1]>0$ and the distribution of $Z_N={\sum }_{i=1}^N{C}_i$ is also IR, we obtain the asymptotics $$P(M_N>x)\sim P(S_N>x)\sim E[\sum _{i=1}^N\bar{F}(x\mathrm{/}{C}_i)]+P(Z_N>x\mathrm{/}E[X_1])\mathrm{.}$$ For completeness, when the distribution of $Z_N$ is IR and heavier than $\bar{F}$, we also obtain conditions under which the asymptotic relations $$P(M_N>x)\sim P(S_N>x)\sim P(Z_N>x\mathrm{/}E[X_1])$$ hold.