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