  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.
