## The Annals of Probability

### A Converse to the Spitzer-Rosen Theorem

Peter Hall

#### 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

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