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December, 1987 A Note on the Variance of a Stopping Time
Robert Keener
Ann. Statist. 15(4): 1709-1712 (December, 1987). DOI: 10.1214/aos/1176350620

Abstract

Let $\{S_n = \sum^n_1 X_i\}_{n\geq 0}$ be a random walk with positive drift $\mu = EX_1 > 0$ and finite variance $\sigma^2 = \operatorname{Var} X_1$. Let $\tau(b) = \inf\{n \geq 1: S_n > b\}, R_b = S_{\tau(b)} - b, M = \min_{n\geq 0} S_n, \tau^+ = \tau(0)$ and $H = S_\tau +$. Lai and Siegmund show that $\operatorname{Var} \tau(b) = b\sigma^2/\mu^3 + K/\mu^2 + o(1)$ as $b \rightarrow \infty$, but give an unpleasant expression for the constant $K$. Using the identity $\int Eh(R_{-y}) dP(M \leq y) = E^+_\tau h(H)/E\tau^+$, the expression for $K$ can be simplified to a form that depends only on moments of ladder variables.

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Robert Keener. "A Note on the Variance of a Stopping Time." Ann. Statist. 15 (4) 1709 - 1712, December, 1987. https://doi.org/10.1214/aos/1176350620

Information

Published: December, 1987
First available in Project Euclid: 12 April 2007

zbMATH: 0637.60058
MathSciNet: MR913584
Digital Object Identifier: 10.1214/aos/1176350620

Subjects:
Primary: 60J15
Secondary: 60G40

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 4 • December, 1987
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