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Probability Theory and Related Fields

, Volume 134, Issue 3, pp 383–416 | Cite as

Exceptional times and invariance for dynamical random walks

  • Davar KhoshnevisanEmail author
  • David A. Levin
  • Pedro J. Méndez-Hernández
Article

Abstract

Consider a sequence Open image in new window of i.i.d. random variables. Associate to each X i (0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X i (t)} t ≥0 (i=1,2,···) whose invariant distribution is the law ν of X 1(0).

Benjamini et al. (2003) introduced the dynamical walk S n (t)=X 1(t)+···+X n (t), and proved among other things that the LIL holds for nS n (t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X i (0)'s are standard normal, the classical integral test is not dynamically stable.

Presently, we study the set of times t when nS n (t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov ɛ-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate.

We prove also that the infinite-dimensional process Open image in new window converges weakly in Open image in new window to the Ornstein–Uhlenbeck process in Open image in new window For this we assume only that the increments have mean zero and variance one.

In addition, we extend a result of Benjamini et al. (2003) by proving that if the X i (0)'s are lattice, mean-zero variance-one, and possess (2+ɛ) finite absolute moments for some ɛ>0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest.

Mathematics Subject Classification (2000)

60J25 60J05 60Fxx 28A78 28C20 

Keywords

Dynamical walks Hausdorff dimension Kolmogorov ɛ-entropy gambler's ruin Upper functions The Ornstein-Uhlenbeck process in Wiener space 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Davar Khoshnevisan
    • 1
    Email author
  • David A. Levin
    • 1
  • Pedro J. Méndez-Hernández
    • 1
    • 2
  1. 1.Department of MathematicsThe University of UtahSalt Lake City
  2. 2.Escuela de MatematicaUniversidad de Costa RicaCosta Rica

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