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Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment
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  • Published: October 1997

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

  • C. Boldrighini1,
  • R. A. Minlos2 &
  • A. Pellegrinotti3 

Probability Theory and Related Fields volume 109, pages 245–273 (1997)Cite this article

  • 137 Accesses

  • 18 Citations

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università di Camerino, via Madonna delle Carceri 9, I-62032 Camerino, Italy, , , , , , IT

    C. Boldrighini

  2. Institute for Problems of Information Transmission, Russian Academy of Sciences, , , , , , RU

    R. A. Minlos

  3. Dipartimento di Matematica, Università degli studi di Roma Tre, Largo S. Leonardo Murialdo 1, I-00146 Roma, Italy, , , , , , IT

    A. Pellegrinotti

Authors
  1. C. Boldrighini
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  2. R. A. Minlos
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  3. A. Pellegrinotti
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Additional information

Received: 4 January 1996/In revised form: 26 May 1997

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Boldrighini, C., Minlos, R. & Pellegrinotti, A. Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment. Probab Theory Relat Fields 109, 245–273 (1997). https://doi.org/10.1007/s004400050132

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  • Issue Date: October 1997

  • DOI: https://doi.org/10.1007/s004400050132

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  • Key words: Random walk
  • Random environment
  • Central limit theorem
  • AMS Subject Classification (1991): 60J15
  • 60F05
  • 60G60
  • 82B41
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