The Annals of Probability

On Stopping Rules and the Expected Supremum of $S_n/a_n$ and $|S_n|/a_n$

Michael J. Klass

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Abstract

Let $\{X_n\}$ be a sequence of i.i.d. mean zero random variables. Let $S_n = X_1 + \cdots + X_n$. This paper is devoted to determining the conditions where-by $E\sup_{n\geqq 1}S_n/a_n < \infty$ and $E\sup_{n\geqq 1}|S_n|/a_n < \infty$ for quite general sequences of increasing constants $\{a_n\}$. For the sequences $\{a_n\}$ considered, we find it sufficient to examine whether or not $\lim_{n\rightarrow\infty} E(\sum^n_{k=1}X_k/a_k)^+ < \infty$. The existence of optimal extended-valued stopping rules with finite expected reward for sequences $\{S_n/a_n\}$ or $\{|S_n|/a_n\}$ is a by-product of our results. This generalizes results of D. L. Burkholder, Burgess Davis, R. F. Gundy, B. J. McCabe and L. A. Shepp, who treat the case $a_n = n$.

Article information

Source
Ann. Probab., Volume 2, Number 5 (1974), 889-905.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996555

Digital Object Identifier
doi:10.1214/aop/1176996555

Mathematical Reviews number (MathSciNet)
MR380967

Zentralblatt MATH identifier
0325.60043

JSTOR
links.jstor.org

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
$S_n/a_n$ stopping rule supremum expected value a.s. convergence

Citation

Klass, Michael J. On Stopping Rules and the Expected Supremum of $S_n/a_n$ and $|S_n|/a_n$. Ann. Probab. 2 (1974), no. 5, 889--905. doi:10.1214/aop/1176996555. https://projecteuclid.org/euclid.aop/1176996555


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