The Annals of Probability

A Converse to the Spitzer-Rosen Theorem

Peter Hall

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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


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