Series expansions for the all-time maximum of α-stable random walks

We study random walks whose increments are a-stable distributions with shape parameter 1 < alpha < 2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an a-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter beta = -1, for the expected value of the all-time maximum of an alpha-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of alpha-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an alpha-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.