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

#### Abstract

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 0 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 cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 744-767.

Dates
First available in Project Euclid: 19 September 2016

https://projecteuclid.org/euclid.aap/1474296313

Mathematical Reviews number (MathSciNet)
MR3568890

Zentralblatt MATH identifier
1351.60054

Subjects
Primary: 60G50: Sums of independent random variables; random walks

#### Citation

Hurvich, Clifford; Reed, Josh. Series expansions for the all-time maximum of α-stable random walks. Adv. in Appl. Probab. 48 (2016), no. 3, 744--767. https://projecteuclid.org/euclid.aap/1474296313

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