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Estimating tails of independently stopped random walks using concave approximations of hazard functions

Part of: Limit theorems Integral transforms, operational calculus Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  16 September 2021

Jaakko Lehtomaa
Jaakko Lehtomaa*
Affiliation:
University of Helsinki
*
*Postal address: Department of Mathematics and Statistics, PO Box 4 (Yliopistonkatu 3), 00014 University of Helsinki, Finland. Email address: [email protected]
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Abstract

This paper considers logarithmic asymptotics of tails of randomly stopped sums. The stopping is assumed to be independent of the underlying random walk. First, finiteness of ordinary moments is revisited. Then the study is expanded to more general asymptotic analysis. Results are applicable to a large class of heavy-tailed random variables. The main result enables one to identify if the asymptotic behaviour of a stopped sum is dominated by its increments or the stopping variable. As a consequence, new sufficient conditions for the moment determinacy of compounded sums are obtained.

Keywords

Heavy-tailedlogarithmic asymptoticsmoment determinacyrandom stoppingStieltjes moment problem

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60E05: Distributions 60F10: Large deviations 44A60: Moment problems
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 773 - 793
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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