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April, 1979 Extended Renewal Theory and Moment Convergence in Anscombe's Theorem
Y. S. Chow, Chao A. Hsiung, T. L. Lai
Ann. Probab. 7(2): 304-318 (April, 1979). DOI: 10.1214/aop/1176995090


In this paper, an $L_p$ analogue of Anscombe's theorem is shown to hold and is then applied to obtain the variance and other central moments of the first passage time $T_c = \inf\{n \geqslant 1 : S_n > cn^\alpha\}$, where $0 \leqslant \alpha < 1, S_n = X_1 + \cdots + X_n$ and $X_1, X_2, \cdots$ are i.i.d. random variables with $EX_1 > 0$. The variance of $T_c$ in the special case $\alpha = 0$ has been studied by various authors in classical renewal theory, and our approach in this paper provides a simple treatment and a natural extension (to the case of a general $\alpha$) of this classical result. The related problem concerning the asymptotic behavior of $\max_{j\leqslant n}j^{-\alpha}S_j$ is also studied, and in this connection, certain maximal inequalities are obtained and they are applied to prove the corresponding moment convergence results of the theorems of Erdos and Kac, and of Teicher.


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Y. S. Chow. Chao A. Hsiung. T. L. Lai. "Extended Renewal Theory and Moment Convergence in Anscombe's Theorem." Ann. Probab. 7 (2) 304 - 318, April, 1979.


Published: April, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0405.60020
MathSciNet: MR525056
Digital Object Identifier: 10.1214/aop/1176995090

Primary: 60F05
Secondary: 60G40 , 60G50 , 60K05

Keywords: Anscombe's theorem , extended renewal theory , Maximal inequalities , moment convergence , uniform integrability , variance of stopping times

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 2 • April, 1979
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