Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment |
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Authors: | C Boldrighini R A Minlos A Pellegrinotti |
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Institution: | (1) Dipartimento di Matematica e Fisica, Università di Camerino, via Madonna delle Carceri 9, I-62032 Camerino, Italy, IT;(2) Institute for Problems of Information Transmission, Russian Academy of Sciences, RU;(3) Dipartimento di Matematica, Università degli studi di Roma Tre, Largo S. Leonardo Murialdo 1, I-00146 Roma, Italy, IT |
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Abstract: | Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ
t
(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X
t
+1= y|X
t
= x,ξ
t
=η) =P
0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ
t
(x):(t,x) ∈ℤν+1} are i.i.d., that both P
0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P
0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X
t
, and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X
t
and a corresponding correction of order to the C.L.T.. Proofs are based on some new L
p
estimates for a class of functionals of the field.
Received: 4 January 1996/In revised form: 26 May 1997 |
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Keywords: | : Random walk Random environment Central limit theorem |
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