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March, 1993 Asymptotic Expansions for the Moments of a Randomly Stopped Average
Girish Aras, Michael Woodroofe
Ann. Statist. 21(1): 503-519 (March, 1993). DOI: 10.1214/aos/1176349039

Abstract

Let $S_1$, $S_2,\cdots$ denote a driftless random walk with values in an inner product space $\mathscr{W}$; let $Z_1$, $Z_2,\cdots$ denote a perturbed random walk of the form $Z_n=n+\langle c,S_n \rangle+\xi_n$, $n = 1, 2,\cdots$, where $\xi_1,\xi_2,\cdots$ are slowly changing, $\langle\centerdot,\centerdot\rangle$ denotes the inner product, and $c\in\mathscr{W}$; and let $t=t_a=inf{n\geq1:Z_n>a}$ for $0\leq a<\infty$. Conditions are developed under which the first four moments of $X_t:=S_t/t$ have asymptotic expansions, and the expansions are found. Stopping times of this form arise naturally in sequential estimation problems, and the main results may be used to find asymptotic expansions for risk functions in such problems. Examples of such applications are included.

Citation

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Girish Aras. Michael Woodroofe. "Asymptotic Expansions for the Moments of a Randomly Stopped Average." Ann. Statist. 21 (1) 503 - 519, March, 1993. https://doi.org/10.1214/aos/1176349039

Information

Published: March, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0788.62075
MathSciNet: MR1212190
Digital Object Identifier: 10.1214/aos/1176349039

Subjects:
Primary: 62L12

Keywords: Anscombe's theorem , Martingales , Maximal inequalities , nonlinear renewal theory , risk functions , sequential estimation , stopping times

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.21 • No. 1 • March, 1993
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