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December, 1971 On the Existence of the Optimal Stopping Rule in the $S_n/n$ Problem When the Second Moment is Infinite
M. E. Thompson, A. K. Basu, W. L. Owen
Ann. Math. Statist. 42(6): 1936-1942 (December, 1971). DOI: 10.1214/aoms/1177693060

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random variables with mean 0, and let $S_n = \sum^n_{i=1} X_i$. The $S_n/n$ optimal stopping problem is to maximize $E(S_\tau/\tau)$ among finite-valued stopping times $\tau$ relative to the process $(S_n, n \geqq 1)$. In this paper we prove partially Dvoretzky's (1967) conjecture that an optimal stopping time should exist when $E|X_1|^\beta < \infty$ for some $\beta > 1$, by showing that the result holds if $\lim \sup_{n\rightarrow\infty} P(S_n \geqq c\|S_n\|) > 0$ for some $c > 0$, where $\|S_n\| = (E|S_n|^\beta)^{1/\beta}$. This condition is shown to hold in some special cases, including the case where the $X_i$ are in the domain of attraction of a stable distribution with exponent greater than one.

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M. E. Thompson. A. K. Basu. W. L. Owen. "On the Existence of the Optimal Stopping Rule in the $S_n/n$ Problem When the Second Moment is Infinite." Ann. Math. Statist. 42 (6) 1936 - 1942, December, 1971. https://doi.org/10.1214/aoms/1177693060

Information

Published: December, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0231.62097
MathSciNet: MR300371
Digital Object Identifier: 10.1214/aoms/1177693060

Rights: Copyright © 1971 Institute of Mathematical Statistics

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Vol.42 • No. 6 • December, 1971
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