Asymptotics for weighted random sums

Abstract

Let {X_i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C₁, C₂,...) 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_iX_i, and its maximum, M_N=sup_{{1≤k N+1}∑_i=1^k C_iX_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[X₁]>0 and the distribution of Z_N=∑ _i=1^NC_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[X₁]). 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[X₁] hold.