Despite a true antipathy to the subject Hardy contributed deeply to modern probability. His work with Ramanujan begat probabilistic number theory. His work on Tauberian theorems and divergent series has probabilistic proofs and interpretations. Finally Hardy spaces are a central ingredient in stochastic calculus. This paper reviews his prejudices and accomplishments through these examples. ∗Dep...

In this paper, we will give a sufficient condition for a non-negative random variable X to be heavy tailed by investigating the Laplace-Stieltjes transform of the probability distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. Our theorem is a kind of Tauberian theorems.

In this paper, a one-sided condition is given to recover .C; ̨/ summability of a sequence from its .A/.C; ̨C 1/ summability. Our result extends and generalizes the well known classical Tauberian theorems given for Abel and Cesàro summability methods. 2010 Mathematics Subject Classification: 40E05; 40G05; 40G10

The aim of this paper is to continue the study asymptotic expansions and summability in a monomial any number variables. In particular we characterize these terms bounded derivatives develop tauberian theorems for processes involved. Furthermore, apply Borel-Laplace analysis framework prove solutions specific class singularly perturbed PDEs.