Moments and Error Rates of Two-Sided Stopping Rules

Abstract

For $X_1, X_2,\cdots$ i.i.d., $EX_1 = \mu \neq 0, S_n = X_1 + \cdots + X_n$, the asymptotic behavior of moments and error rates of the two-sided stopping rules $\inf \{n \geq 1: |S_n| > cn^\alpha\}, c > 0, 0 \leq \alpha < 1$, is considered. Convergence of (normalized) moments of all orders as $c \rightarrow \infty$ is obtained, without the higher moment assumptions needed in the one-sided case of extended renewal theory (Gut, 1974), and in a more general setting than just the i.i.d. case. Necessary and sufficient conditions are given for convergence of series involving the error rates, in terms of the moments of $X_1$.