The Annals of Statistics

A Note on the Variance of a Stopping Time

Robert Keener

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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.

Article information

Source
Ann. Statist., Volume 15, Number 4 (1987), 1709-1712.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350620

Digital Object Identifier
doi:10.1214/aos/1176350620

Mathematical Reviews number (MathSciNet)
MR913584

Zentralblatt MATH identifier
0637.60058

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
Random walks ladder variables stopping times excess over the boundary

Citation

Keener, Robert. A Note on the Variance of a Stopping Time. Ann. Statist. 15 (1987), no. 4, 1709--1712. doi:10.1214/aos/1176350620. https://projecteuclid.org/euclid.aos/1176350620


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