## The Annals of Probability

### The Class of Limit Laws for Stochastically Compact Normed Sums

William E. Pruitt

#### Abstract

Khintchine showed that every infinitely divisible law can be obtained as the limit of a subsequence of normed sums of independent, identically distributed random variables. Here we restrict the summands to be in a class which makes the normed sums stochastically compact, i.e. so that every subsequence has a further subsequence which converges to a nondegenerate limit. A nice analytic condition for stochastic compactness was obtained by Feller. Our result is an analogous characterization of the class of limit laws of subsequences of stochastically compact normed sums. One consequence is that they have $C^\infty$ densities.

#### Article information

Source
Ann. Probab., Volume 11, Number 4 (1983), 962-969.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993445

Digital Object Identifier
doi:10.1214/aop/1176993445

Mathematical Reviews number (MathSciNet)
MR714959

Zentralblatt MATH identifier
0519.60014

JSTOR