Advertisement

Explicit Laws for the Records of the Perturbed Random Walk on \(\mathbb {Z}\)

  • Laurent Serlet
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

Abstract

We consider a nearest neighbor random walk on \(\mathbb {Z}\) which is perturbed when it reaches its extrema, as considered before by several authors. We give invariance principles for the signs of the records, the values of the records, the times of the records, the number of visited points, with explicit asymptotic Laplace transforms and/or densities.

Keywords

Perturbed random walk Once-reinforced random walk Perturbed Brownian motion Records Invariance principle Recurrence 

References

  1. 1.
    P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968)zbMATHGoogle Scholar
  2. 2.
    P. Carmona, F. Petit, M. Yor, Beta variables as times spent in [0,  +) by certain perturbed Brownian motions. J. Lond. Math. Soc. 58, 239–256 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Chaumont, R.A. Doney, Y. Hu, Upper and lower limits of doubly perturbed Brownian motion. Ann. Inst. H. Poincaré Probab. Stat. 36(2), 219–249 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Davis, Weak limits of perturbed random walks and the equation \(Y_t=B_t+\alpha \sup \{Y_s:\;s\leq t\}+\beta \inf \{Y_s;\;s\leq t\}\). Ann. Probab. 24(4), 2007–2023 (1996)Google Scholar
  5. 5.
    B. Davis, Brownian motion and random walk perturbed at extrema. Probab. Theory Relat. Fields 113, 501–518 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    S.N. Ethier, T.G. Kurtz, Markov Processes; Characterization and Convergence (Wiley, New York, 2009)zbMATHGoogle Scholar
  7. 7.
    R. Pemantle, A survey of random processes with reinforcement. Probab. Surv. 4, 1–79 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Perman, W. Werner, Perturbed Brownian motions. Probab. Theory Relat. Fields 108, 357–383 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Serlet, Hitting times for the perturbed reflecting random walk. Stoch. Process. Appl. 123, 110–130 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    W. Werner, Some remarks on perturbed reflecting Brownian motion, in Séminaire de Probabilités XXIX. Lecture Notes in Mathematics, vol. 1613 (Springer, Berlin, 1995), pp. 37–43Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Blaise Pascal (CNRS UMR 6620)Université Clermont AuvergneAubièreFrance

Personalised recommendations