Asymptotics for Weighted Random Sums
Let {X i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C 1, C 2,…) be a nonnegative random vector independent of the {X i } with N∈ℕ∪ {∞}. We study the weighted random sum S N =∑{i=1} N C i X i , and its maximum, M N =sup{1≤k N+1∑ i=1 k C i X i . This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄(x/C i )] as x→∞. When E[X 1]>0 and the distribution of Z N =∑ i=1 N C i is also intermediate regularly varying, we obtain the asymptotics P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄}(x/C i )] +P(Z N > x/E[X 1]). For completeness, when the distribution of Z N is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(M N > x) ∼ P(S N > x)∼ P(Z N > x / E[X 1 ] hold.