On the Maximum of a Random Walk with Small Negative Drift

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. mean zero random variables. Let $S_n = X_1 + \cdots + X_n$ and $M_\varepsilon = \sup_{n\geqq 1} (S_n - n\varepsilon)^+$ for $\varepsilon > 0$. Suppose $\sigma^2 = E(X_1)^2$ is positive and finite. Then $EM_\varepsilon < \infty$ and $2\varepsilon\sigma^{-2} EM_\varepsilon$ converges to 1 as $\varepsilon \searrow 0^+$. In this paper we obtain an approximation of the discrepancy $1 - 2\varepsilon\sigma^{-2} EM_\varepsilon$ as $\varepsilon \searrow 0^+$. To do so we derive a first order approximation of $P(M_\varepsilon < y)$ which is uniform in $y$ as $\varepsilon \searrow 0^+$ and asymptotically exact for $y$ on $\lbrack y_\varepsilon, \infty)$ provided $y_\varepsilon \rightarrow \infty$. Approximation of $P(M_\varepsilon < y)$ necessitates a digression into renewal theory. We derive an approximation of the expected time $E\tau_y$ required by a sum $T_n = Y_1 + \cdots + Y_n$ of i.i.d. non-negative random variables to reach or exceed $y$. The bounds obtained are of particular interest when $EX = \infty$ and are best possible in a rather strong sense.