Journal of Theoretical Probability

, Volume 28, Issue 1, pp 1–40 | Cite as

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

  • Gerold Alsmeyer
  • Alexander Iksanov
  • Matthias Meiners


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.


First passage time Last exit time Number of visits Perturbed random walk Random walk Renewal theory Shot-noise process 

Mathematics Subject Classification (2010)

60G50 60G40 



The research of Gerold Alsmeyer was supported by DFG SFB 878 “Geometry, Groups and Actions”. A part of this study was done while Alexander Iksanov was visiting Münster in January/February and May 2011. Iksanov acknowledges the financial support and hospitality and also supported by a Grant awarded by the President of Ukraine (project \(\Phi \)47/012) and partly supported by a Grant from Utrecht University, the Netherlands. Research of Matthias Meiners was partly supported by DFG-grant Me 3625/1-1 and DFG SFB 878 “Geometry, Groups and Actions”. The authors thank an anonymous referee for a careful reading of the manuscript and helpful comments.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Gerold Alsmeyer
    • 1
  • Alexander Iksanov
    • 2
  • Matthias Meiners
    • 3
  1. 1.Institut für Mathematische StatistikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Faculty of CyberneticsNational T. Shevchenko University of KievKievUkraine
  3. 3.Institut für Mathematische StatistikWestfälische Wilhelms-Universität MünsterMünsterGermany

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