The Annals of Probability

Extended Renewal Theory and Moment Convergence in Anscombe's Theorem

Y. S. Chow, Chao A. Hsiung, and T. L. Lai

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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.

Article information

Ann. Probab. Volume 7, Number 2 (1979), 304-318.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G50: Sums of independent random variables; random walks 60K05: Renewal theory

Extended renewal theory variance of stopping times Anscombe's theorem moment convergence uniform integrability maximal inequalities


Chow, Y. S.; Hsiung, Chao A.; Lai, T. L. Extended Renewal Theory and Moment Convergence in Anscombe's Theorem. Ann. Probab. 7 (1979), no. 2, 304--318. doi:10.1214/aop/1176995090.

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