Power and Exponential Moments of the Number of Visits and Related Quantities for Perturbed Random Walks

Gerold Alsmeyer (), Alexander Iksanov () and Matthias Meiners ()
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Gerold Alsmeyer: Westfälische Wilhelms-Universität Münster
Alexander Iksanov: National T. Shevchenko University of Kiev
Matthias Meiners: Westfälische Wilhelms-Universität Münster

Journal of Theoretical Probability, 2015, vol. 28, issue 1, 1-40

Abstract: Abstract Let $$(\xi _1,\eta _1),(\xi _2,\eta _2),\ldots $$ be a sequence of i.i.d. copies of a random vector $$(\xi ,\eta )$$ taking values in $$\mathbb{R }^2$$ , and let $$S_n:= \xi _1+\cdots +\xi _n$$ . The sequence $$(S_{n-1} + \eta _n)_{n \ge 1}$$ is then called perturbed random walk. We study random quantities defined in terms of the perturbed random walk: $$\tau (x)$$ , the first time the perturbed random walk exits the interval $$(-\infty ,x]; \,N(x)$$ , the number of visits to the interval $$(-\infty ,x]$$ ; and $$\rho (x)$$ , the last time the perturbed random walk visits the interval $$(-\infty ,x]$$ . We provide criteria for the almost sure finiteness and for the finiteness of exponential moments of these quantities. Further, we provide criteria for the finiteness of power moments of $$N(x)$$ and $$\rho (x)$$ . In the course of the proofs of our main results, we investigate the finiteness of power and exponential moments of shot-noise processes and provide complete criteria for both, power and exponential moments.

Keywords: First passage time; Last exit time; Number of visits; Perturbed random walk; Random walk; Renewal theory; Shot-noise process; 60G50; 60G40 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-012-0475-7

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