The Annals of Statistics

Asymptotic Expansions for the Moments of a Randomly Stopped Average

Girish Aras and Michael Woodroofe

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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.

Article information

Source
Ann. Statist., Volume 21, Number 1 (1993), 503-519.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349039

Digital Object Identifier
doi:10.1214/aos/1176349039

Mathematical Reviews number (MathSciNet)
MR1212190

Zentralblatt MATH identifier
0788.62075

JSTOR
links.jstor.org

Subjects
Primary: 62L12: Sequential estimation

Keywords
Anscombe's theorem martingales maximal inequalities nonlinear renewal theory sequential estimation stopping times risk functions

Citation

Aras, Girish; Woodroofe, Michael. Asymptotic Expansions for the Moments of a Randomly Stopped Average. Ann. Statist. 21 (1993), no. 1, 503--519. doi:10.1214/aos/1176349039. https://projecteuclid.org/euclid.aos/1176349039


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