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

Part of: Distribution theory - Probability Distribution theory Qualitative behavior

Published online by Cambridge University Press:  26 July 2018

Dmitri Finkelshtein  and
Pasha Tkachov
Dmitri Finkelshtein*
Affiliation:
Swansea University
Pasha Tkachov*
Affiliation:
Universität Bielefeld
*
* Postal address: Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK. Email address: [email protected]
** Current address: Gran Sasso Science Institute, Viale Francesco Crispi, 7, 67100 L'Aquila AQ, Italy. Email address: [email protected]
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Abstract

We study the tail asymptotic of subexponential probability densities on the real line. Namely, we show that the n-fold convolution of a subexponential probability density on the real line is asymptotically equivalent to this density multiplied by 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 subexponential functions and use it to find an analogue of Kesten's bound for functions on ℝd. The results are applied to the study of the fundamental solution to a nonlocal heat equation.

Keywords

Subexponential densitylong-tail functionconvolution tailheavy-tailed distributiontail-equivalenceasymptotic behavior

MSC classification

Primary: 60E05: Distributions
Secondary: 45M05: Asymptotics 62E20: Asymptotic distribution theory
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 373 - 395
Copyright
Copyright © Applied Probability Trust 2018 

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