Cumulants of the maximum of the Gaussian random walk

Abstract

Let X 1 , X 2 , … be independent variables, each having a normal distribution with negative mean − β < 0 and variance 1. We consider the partial sums S n = X 1 + ⋯ + X n , with S 0 = 0 , and refer to the process { S n : n ≥ 0 } as the Gaussian random walk. This paper is concerned with the cumulants of the maximum M β = max { S n : n ≥ 0 } . We express all cumulants of M β in terms of Taylor series about β at 0 with coefficients that involve the Riemann zeta function. Building upon the work of Chang and Peres [J.T. Chang, Y. Peres, Ladder heights, Gaussian random walks and the Riemann zeta function, Ann. Probab. 25 (1997) 787–802] on P ( M β = 0 ) and Bateman’s formulas on Lerch’s transcendent, expressions of this type for the first and second cumulants of M β have been previously obtained by the authors [A.J.E.M. Janssen, J.S.H. van Leeuwaarden, On Lerch’s transcendent and the Gaussian random walk, Ann. Appl. Probab. 17 (2007) 421–439]. The method is systemized in this paper to yield similar Taylor series expressions for all cumulants. The key idea in obtaining the Taylor series for the k th cumulant is to differentiate its Spitzer-type expression (involving the normal distribution) k + 1 times, rewrite the resulting expression in terms of Lerch’s transcendent, and integrate k + 1 times. The major issue then is to determine the k + 1 integration constants, for which we invoke Euler–Maclaurin summation, among other things. Since the Taylor series are only valid for β < 2 π , we obtain alternative series expansions that can be evaluated for all β > 0 . We further present sharp bounds on P ( M β = 0 ) and the first two moments of M β . We show how the results in this paper might find important applications, particularly for queues in heavy traffic, the limiting overshoot in boundary crossing problems and the equidistant sampling of Brownian motion.