Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

• Published in: Probability theory and related fields Vol. 109; no. 2; pp. 245 - 273
• Main Authors: Boldrighini, C., Minlos, R. A., Pellegrinotti, A.
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.10.1997
• Subjects: Central limit theorem; Covariance matrix; Markov chains; Mathematics; Probability; Random walk; Theorems; Transition probabilities
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s004400050132

Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξt(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(Xt+1= y|Xt= x,ξt=η) =P0( y−x)+ c(y−x;η(x)). We assume that the variables {ξt(x):(t,x) ∈ℤν+1} are i.i.d., that both P0(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 P0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of Xt, 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 Xt and a corresponding correction of order to the C.L.T.. Proofs are based on some new Lp estimates for a class of functionals of the field.