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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
Article

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 

Mathematics Subject Classification (2010)

60G50 60G40 

Notes

Acknowledgments

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.

References

  1. 1.
    Alsmeyer, G.: Erneuerungstheorie. Teubner Skripten zur Mathematischen Stochastik. [Teubner texts on Mathematical Stochastics]. B. G. Teubner, Stuttgart (1991). Analyse stochastischer regenerationsschemata. [Analysis of stochastic regeneration schemes]Google Scholar
  2. 2.
    Alsmeyer, G.: On generalized renewal measures and certain first passage times. Ann. Probab. 20(3), 1229–1247 (1992)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Alsmeyer, G., Iksanov, A.: A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks. Electron. J. Probab. 14(10), 289–312 (2009)MATHMathSciNetGoogle Scholar
  4. 4.
    Alsmeyer, G., Iksanov, A., Rösler, U.: On distributional properties of perpetuities. J. Theoret. Probab. 22(3), 666–682 (2009)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Araman, V.F., Glynn, P.W.: Tail asymptotics for the maximum of perturbed random walk. Ann. Appl. Probab. 16(3), 1411–1431 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)MATHGoogle Scholar
  7. 7.
    Burkholder, D.L., Davis, B.J., Gundy R.F.: Integral inequalities for convex functions of operators on martingales. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, CA, 1970/1971), vol. II: Probability Theory, pp. 223–240. University of California Press, Berkeley, CA (1972)Google Scholar
  8. 8.
    Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304 (1970)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chow, Y.S.: On the moments of ladder epochs for driftless random walks. J. Appl. Probab. 31A, 201–205 (1994). Studies in applied probabilityCrossRefGoogle Scholar
  10. 10.
    Chow, Y.S., Robbins, H., Siegmund, D.: The Theory of Optimal Stopping. Dover Publications Inc., New York (1991) Corrected reprint of the 1971 originalGoogle Scholar
  11. 11.
    Doney, R.A., O’Brien, G.L.: Loud shot noise. Ann. Appl. Probab. 1(1), 88–103 (1991)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Erickson, K.B.: The strong law of large numbers when the mean is undefined. Trans. Am. Math. Soc. 185, 371–381 (1973)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fill, J.A., Huber, M.L.: Perfect simulation of Vervaat perpetuities. Electron. J. Probab. 15, 96–109 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Gnedin, A., Iksanov, A., Marynych, A.: The Bernoulli sieve: an overview. Discrete Math. Theor. Comput. Sci. Proc., AI, pages 329–342. Assoc. Discrete Math. Theor. Comput. Sci. Nancy. Springer, London (2010)Google Scholar
  15. 15.
    Gnedin, A., Iksanov, A., Marynych, A.: Limit theorems for the number of occupied boxes in the Bernoulli sieve. Theory Stoch. Process. 16(32)(2), 44–57 (2010)Google Scholar
  16. 16.
    Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126–166 (1991)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Goldie, C.M., Maller, R.A.: Stability of perpetuities. Ann. Probab. 28(3), 1195–1218 (2000)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Gut, A.: On the moments and limit distributions of some first passage times. Ann. Probab. 2, 277–308 (1974)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Gut, A.: Stopped random walks. Limit theorems and applications. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2009)Google Scholar
  20. 20.
    Hao, X., Tang, Q., Wei, L.: On the maximum exceedance of a sequence of random variables over a renewal threshold. J. Appl. Probab. 46(2), 559–570 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hitczenko, P.: Comparison of moments for tangent sequences of random variables. Probab. Theory Related Fields 78(2), 223–230 (1988)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hitczenko, P.: On tails of perpetuities. J. Appl. Probab. 47(4), 1191–1194 (2010)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Hitczenko, P., Wesołowski, J.: Perpetuities with thin tails revisited. Ann. Appl. Probab. 19(6), 2080–2101 (2009) Corrigendum in 20(3):1177 (2010)Google Scholar
  24. 24.
    Hitczenko, P., Wesołowski, J.: Renorming divergent perpetuities. Bernoulli 17(3), 880–894 (2011)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Iksanov, A.: Fixed points of inhomogeneous smoothing transforms. Unpublished manuscript, 2007. Thesis (habilitation)-National T. Shevchenko University of Kiev, KievGoogle Scholar
  26. 26.
    Iksanov, A., Meiners, M.: Exponential moments of first passage times and related quantities for random walks. Electron. Commun. Probab. 15, 365–375 (2010)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Iksanov, A., Meiners, M.: Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks. J. Appl. Probab. 47(2), 513–525 (2010)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Iksanov, A.M.: Parameter estimation for the radioactive contamination process. Studia Sci. Math. Hungar. 37(3–4), 237–258 (2001)MATHMathSciNetGoogle Scholar
  29. 29.
    Iksanov, A.M.: Functional limit theorems for renewal shot noise processes, 2012. Preprint available at arxiv.org/abs/1202.1950Google Scholar
  30. 30.
    Janson, S.: Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift. Adv. Appl. Probab. 18(4), 865–879 (1986)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Kesten, H., Maller, R.A.: Two renewal theorems for general random walks tending to infinity. Probab. Theory Related Fields 106(1), 1–38 (1996)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Konstantopoulos, T., Lin, S.-J.: Macroscopic models for long-range dependent network traffic. Queueing Syst. Theory Appl. 28(1–3), 215–243 (1998)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Lai, T.L., Siegmund, D.: A nonlinear renewal theory with applications to sequential analysis. I. Ann. Statist. 5(5), 946–954 (1977)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Lai, T.L., Siegmund, D.: A nonlinear renewal theory with applications to sequential analysis. II. Ann. Statist. 7(1), 60–76 (1979)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Lebedev, A.V.: Extremes of subexponential shot noise. Math. Notes 71(1–2), 206–210 (2002)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    McCormick, W.P.: Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Probab. 34(3), 643–656 (1997)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Mikosch, T., Resnick, S.: Activity rates with very heavy tails. Stoch. Process. Appl. 116(2), 131–155 (2006)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Palmowski, Z., Zwart, B.: Tail asymptotics of the supremum of a regenerative process. J. Appl. Probab. 44(2), 349–365 (2007)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Palmowski, Z., Zwart, B.: On perturbed random walks. J. Appl. Probab. 47(4), 1203–1204 (2010)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Rice, J.: On generalized shot noise. Adv. Appl. Probab. 9(3), 553–565 (1977)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Schottky, W.: Spontaneous current fluctuations in electron streams. Ann. Phys. 57, 541–567 (1918)CrossRefGoogle Scholar
  42. 42.
    Takács, L.: On secondary stochastic processes generated by recurrent processes. Acta Math. Acad. Sci. Hungar. 7, 17–29 (1956)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Uchiyama, K.: A note on summability of ladder heights and the distributions of ladder epochs for random walk. Stoch. Process. Appl. 121(9), 1938–1961 (2011)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Woodroofe, M.: Nonlinear renewal theory in sequential analysis volume 39 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1982)Google Scholar

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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