Abstract
Let {Xi} 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 {Xi} with N∈ℕ∪ {∞}. We study the weighted random sum SN =∑{i=1}N CiXi, and its maximum, MN=sup{1≤k N+1∑i=1k CiXi. 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(MN > x)∼ P(SN > x)∼ E[∑i=1N F̄(x/Ci)] as x→∞. When E[X1]>0 and the distribution of ZN=∑ i=1NCi is also intermediate regularly varying, we obtain the asymptotics P(MN > x)∼ P(SN > x)∼ E[∑i=1N F̄}(x/Ci)] +P(ZN > x/E[X1]). For completeness, when the distribution of ZN is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(MN > x) ∼ P(SN > x)∼ P(ZN > x / E[X1] hold.
Citation
MARIANA OLVERA-CRAVIOTO. "Asymptotics for weighted random sums." Adv. in Appl. Probab. 44 (4) 1142 - 1172, December 2012. https://doi.org/10.1239/aap/1354716592
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