Statistica Sinica 9(1999), 725-733

A NOTE ON FIRST PASSAGE TIMES

OF STATIONARY SEQUENCES

Hwai-Chung Ho

Academia Sinica, Taipei

Abstract: Let $G(\cdot)$ be a Borel function applied to a stationary, possibly long-memory, sequence of standard Gaussian random variables $\{ X_i \}$. Define the first passage time $T(c) = \inf \{ n \geq 1, S_n \geq c \}, c >0$, for partial sums $S_n=\sum_{i=1}^n G(X_i)$. Suppose $G(X_i)$ has finite positive mean $\mu$. When $G(X_i)$ itself is positive or its negative part is under some moment conditions, it is proved that $E(T(c)/c)^{\gamma} \rightarrow \mu^{-\gamma}$ for $\gamma >0$ as $c$ tends to infinity.

Key words and phrases: Elementary renewal theorem, first passage time, Gaussian sequence, long-memory, long-range dependence, self-similar.