Stochastic compactness and point processes

We show that stochastic compactness of partial sums with no normal limit distribution corresponds to stochastic compactness of the point processes generated by the observations so that there exist joint limit distributions for the sample sums and the sample maxima.

Some Properties of stochastic compactness

The aim of this paper is to show that some of the known properties of distributions in the domain of attraction of a stable law have counterparts for distributions which are stochastically compact in the sense of Feller. This enables us to unify the ideas of Feller and Doeblin, who first studied the concept of stochastic compactness, and give new characterizations of stochastic compactness and the domain of attraction of the normal distribution.

Relative stability, characteristic functions and stochastic compactness

A recent result of Rogozin on the relative stability of a distribution function is extended, by giving equivalences for relative stability in terms of truncated moments of the distribution and in terms of the real and imaginary parts of the characteristic function. As an application, the known results on centering distributions in the domain of attraction of a stable law are extended to the case of stochastically compact distributions.