Series expansions for the all-time maximum of \(\alpha\)-stable random walks. (English) Zbl 1351.60054

Hurvich, Clifford; Reed, Josh

Adv. Appl. Probab. 48, No. 3, 744-767 (2016).

Summary: We study random walks whose increments are \(\alpha\)-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 \(\alpha\)-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.

MSC:

60G50	Sums of independent random variables; random walks
60G52	Stable stochastic processes
60G70	Extreme value theory; extremal stochastic processes
60K25	Queueing theory (aspects of probability theory)
90B22	Queues and service in operations research

Keywords:

\(\alpha\)-stable random walks; maximum; series expansions; doubly stochastic process; queueing theory