Applied Probability Trust (26 August 2015) SERIES EXPANSIONS FOR THE ALL-TIME MAXIMUM OF α- STABLE RANDOM WALKS
نویسندگان
چکیده
We study random walks whose increments are α-stable distributions with shape parameter 1 < α < 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 α-stable random walk is equal to zero and, in the totally skewed to the left case of skewness parameter β = −1, for the expected value of the all-time maximum of an α-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 α-stable random variables and Zolotarev’s integral representation for the CDF of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.
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