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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 {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.

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

Information

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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