On Dynamical Gaussian Random Walks
نویسنده
چکیده
Motivated by the recent work of Benjamini, Häggström, Peres, and Steif (2003) on dynamical random walks, we: (i) Prove that, after a suitable normalization, the dynamical Gaussian walk converges weakly to the Ornstein–Uhlenbeck process in classical Wiener space; (ii) derive sharp tailasymptotics for the probabilities of large deviations of the said dynamical walk; and (iii) characterize (by way of an integral test) the minimal envelop(es) for the growth-rate of the dynamical Gaussian walk. This development also implies the tail capacity-estimates of Mountford (1992) for large deviations in classical Wiener space. The results of this paper give a partial affirmative answer to the problem, raised in Benjamini et al. (2003, Question 4) of whether there are precise connections between the OU process in classical Wiener space and dynamical random walks.
منابع مشابه
SECOND-ORDER FLUCTUATIONS AND CURRENT ACROSS CHARACTERISTIC FOR A ONE-DIMENSIONAL GROWTH MODEL OF INDEPENDENT RANDOM WALKS1 BY TIMO SEPPÄLÄINEN University of Wisconsin
Fluctuations from a hydrodynamic limit of a one-dimensional asymmetric system come at two levels. On the central limit scale n1/2 one sees initial fluctuations transported along characteristics and no dynamical noise. The second order of fluctuations comes from the particle current across the characteristic. For a system made up of independent random walks we show that the second-order fluctuat...
متن کاملExceptional Times for the Dynamical Discrete Web
The dynamical discrete web (DyDW), introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter τ . The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed τ . In this paper, we stu...
متن کاملCentral Limit Theorems for Open Quantum Random Walks∗
Open Quantum Random Walks, as developed in [1], are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random...
متن کاملRandom Walks Derived from Billiards
We introduce a class of random dynamical systems derived from billiard maps, which we call random billiards, and study certain random walks on the real line obtained from them. The interplay between the billiard geometry and the stochastic properties of the random billiard is investigated. Our main results are concerned with the description of the spectrum of the random billiard’s Markov operat...
متن کاملGaussian fluctuations for random walks in random mixing environments
We consider a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions. Continuing our previous work [2] for the law of large numbers, we prove here that the fluctuations are gaussian when the environment is Gibbsian satisfying the “strong mixing condition” of Dobrushin and Shlosman and the mixing rate is large enough...
متن کامل