## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

**Keywords**

Infinitely divisible laws weak convergence class $\mathscr{L}$

#### Citation

Pruitt, William E. The Class of Limit Laws for Stochastically Compact Normed Sums. Ann. Probab. 11 (1983), no. 4, 962--969. doi:10.1214/aop/1176993445. https://projecteuclid.org/euclid.aop/1176993445