Translator Disclaimer
December 2012 Asymptotics for weighted random sums
MARIANA OLVERA-CRAVIOTO
Author Affiliations +
Adv. in Appl. Probab. 44(4): 1142-1172 (December 2012). DOI: 10.1239/aap/1354716592

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+1i=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

Download 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

Information

Published: December 2012
First available in Project Euclid: 5 December 2012

zbMATH: 1263.60042
MathSciNet: MR3052852
Digital Object Identifier: 10.1239/aap/1354716592

Subjects:
Primary: 60G50
Secondary: 60F10 , 60G70 , 60J80

Keywords: Breiman's theorem , heavy tail , intermediate regular variation , randomly stopped sum , Randomly weighted sum , regular variation

Rights: Copyright © 2012 Applied Probability Trust

JOURNAL ARTICLE
31 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.44 • No. 4 • December 2012
Back to Top