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

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 ${\mathbb{R}}^d$. The results are applied for the study of the fundamental solution to a nonlocal heat-equation.

The paper contains materials which were previously included to the first version of the publication arXiv:1611.09329 [math.AP]. A condensed version is to appear in Adv.Appl.Prob