Extended Renewal Theory and Moment Convergence in Anscombe's Theorem

Abstract

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.