First Exit Time of a Random Walk from the Bounds $f(n) \pm cg(n)$, with Applications

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. real-valued random variables with $EX_1 = 0, EX^2_1 < \infty$, and $S_n = X_1 + \cdots + X_n, n = 1, 2, \cdots$. For a chosen positive integer $m$ and real $c > 0$ the exit time $N_c$ is the least integer $n \geqslant m$ such that $f(n) - cg(n) < S_n < f(n) + cg(n)$ is violated, where the functions $f$ and $g (0 < g\uparrow\infty)$ are both defined for all real $x \geqslant m$. Under certain conditions on $f$ and $g$, a function $\psi$ (unique up to an asymptotic equivalence), satisfying $\psi(x) / x \rightarrow 0$ as $x\rightarrow\infty$, is constructed on $\lbrack m, \infty)$ such that $\psi(N_c)$ is exactly exponentially bounded. This result generalizes earlier theorems of Breiman; Chow, Robbins, and Teicher; Gundy and Siegmund; Brown; and Lai. A consequence is that $N_c$ itself is not exponentially bounded. In a multivariate generalization the $X$'s take their values in $R^d$ and $N_c$ is the first exit time of $L_n$ from $(- l(c), l(c))$, where $L_n = n\Phi(S_n/n) - h(n)$, and certain conditions are imposed on $\Phi$ and $h$. Here $\psi(x) = \int^x_mh(t)t^{-1}dt$. The results are applied to show, both in the sequential $F$-test and in the Savage-Sethuraman sequential rank-order test, that for certain distributions of the $X$'s the stopping time is not exponentially bounded.