## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 4 (1981), 633-641.

### A Converse to the Spitzer-Rosen Theorem

#### Abstract

Let $S_n$ be the sum of $n$ independent and identically distributed random variables with zero means and unit variances. The central limit theorem implies that $P(S_n \leq 0) \rightarrow 1/2$, and the Spitzer-Rosen theorem (with refinements by Baum and Katz, Heyde, and Koopmans) provides a rate of convergence in this limit law. In the present paper we investigate the converse of this result. Given a certain rate of convergence of $P(S_n \leq 0)$ to 1/2, what does this imply about the common distribution of the summands?

#### Article information

**Source**

Ann. Probab., Volume 9, Number 4 (1981), 633-641.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994368

**Digital Object Identifier**

doi:10.1214/aop/1176994368

**Mathematical Reviews number (MathSciNet)**

MR624689

**Zentralblatt MATH identifier**

0462.60025

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G50: Sums of independent random variables; random walks

**Keywords**

Central limit theorem rate of convergence Spitzer-Rosen theorem sum of independent random variables

#### Citation

Hall, Peter. A Converse to the Spitzer-Rosen Theorem. Ann. Probab. 9 (1981), no. 4, 633--641. doi:10.1214/aop/1176994368. https://projecteuclid.org/euclid.aop/1176994368