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August, 1981 A Converse to the Spitzer-Rosen Theorem
Peter Hall
Ann. Probab. 9(4): 633-641 (August, 1981). DOI: 10.1214/aop/1176994368


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?


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Peter Hall. "A Converse to the Spitzer-Rosen Theorem." Ann. Probab. 9 (4) 633 - 641, August, 1981.


Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60025
MathSciNet: MR624689
Digital Object Identifier: 10.1214/aop/1176994368

Primary: 60F05
Secondary: 60G50

Keywords: central limit theorem , rate of convergence , Spitzer-Rosen theorem , Sum of independent random variables

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 4 • August, 1981
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