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We study the tail asymptotic of sub-exponential probability densities on thereal line. Namely, we show that the n-fold convolution of a sub-exponentialprobability density on the real line is asymptotically equivalent to thisdensity times n. We prove Kesten's bound, which gives a uniform in n estimateof the n-fold convolution by the tail of the density. We also introduce a classof regular sub-exponential functions and use it to find an analogue of Kesten'sbound for functions on $\mathbb{R}^d$. The results are applied for the study ofthe fundamental solution to a nonlocal heat-equation.

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