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Asymptotics for Weighted Random Sums

Part of: Stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  04 January 2016

Mariana Olvera-Cravioto
Mariana Olvera-Cravioto*
Affiliation:
Columbia University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. Email address: molvera@ieor.columbia.edu
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Abstract

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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}NCiXi, and its maximum, MN=sup{1≤kN+1∑i=1kCiXi. 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=1NF̄(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=1NF̄}(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.

Keywords

Randomly weighted sumrandomly stopped sumheavy tailintermediate regular variationregular variationBreiman's theorem

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F10: Large deviations 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 1142 - 1172
Copyright
© Applied Probability Trust 

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