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Kesten's bound for subexponential densities on the real line and its multi-dimensional analogues

Dmitri Finkelshtein Orcid Logo, Pasha Tkachov

Advances in Applied Probability, Volume: 50, Issue: 02, Pages: 373 - 395

Swansea University Author: Dmitri Finkelshtein Orcid Logo

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DOI (Published version): 10.1017/apr.2018.18

Abstract

We study the tail asymptotic of sub-exponential probability densities on the real line. Namely, we show that the n-fold convolution of a sub-exponential probability density on the real line is asymptotically equivalent to this density times n. We prove Kesten's bound, which gives a uniform in n...

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Published in: Advances in Applied Probability
ISSN: 0001-8678 1475-6064
Published: Applied Probability Trust/Cambridge University Press 2018
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URI: https://cronfa.swan.ac.uk/Record/cronfa38336
Abstract: We study the tail asymptotic of sub-exponential probability densities on the real line. Namely, we show that the n-fold convolution of a sub-exponential probability density on the real line is asymptotically equivalent to this density times n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular sub-exponential functions and use it to find an analogue of Kesten's bound for functions on R^d. The results are applied for the study of the fundamental solution to a nonlocal heat-equation.
Keywords: sub-exponential densities; long-tail functions; heavy-tailed distributions; convolution tails; tail-equivalence; asymptotic behavior
College: Faculty of Science and Engineering
Issue: 02
Start Page: 373
End Page: 395

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