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Exceptional times and invariance for dynamical random walks
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  • Published: 03 May 2005

Exceptional times and invariance for dynamical random walks

  • Davar Khoshnevisan1,
  • David A. Levin1 &
  • Pedro J. Méndez-Hernández1,2 

Probability Theory and Related Fields volume 134, pages 383–416 (2006)Cite this article

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  • 7 Citations

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Abstract

Consider a sequence 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 n↦S 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 n↦S 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 converges weakly in to the Ornstein–Uhlenbeck process in 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, UT, 84112–0090

    Davar Khoshnevisan, David A. Levin & Pedro J. Méndez-Hernández

  2. Escuela de Matematica, Universidad de Costa Rica, San Pedro de Montes de Oca, Costa Rica

    Pedro J. Méndez-Hernández

Authors
  1. Davar Khoshnevisan
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  2. David A. Levin
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  3. Pedro J. Méndez-Hernández
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Corresponding author

Correspondence to Davar Khoshnevisan.

Additional information

The research of D. Kh. is partially supported by a grant from the NSF.

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Khoshnevisan, D., Levin, D. & Méndez-Hernández, P. Exceptional times and invariance for dynamical random walks. Probab. Theory Relat. Fields 134, 383–416 (2006). https://doi.org/10.1007/s00440-005-0435-6

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  • Received: 30 September 2004

  • Revised: 28 January 2005

  • Published: 03 May 2005

  • Issue Date: March 2006

  • DOI: https://doi.org/10.1007/s00440-005-0435-6

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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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