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Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$


Abstract

In many models of Mathematical Physics, based on the study of a Markov chain $\widehat \eta= \{\eta_{t}\}_{t=0}^{\infty}$ on ${\mathbb Z}^d$, one can prove by perturbative arguments a contraction property of the stochastic operator restricted to a subspace of local functions $\mathcal H_{M}$ endowed with a suitable norm. We show, on the example of a model of random walk in random environment with mutual interaction, that the condition is enough to prove a Central Limit Theorem for sequences $\{f(S^{k}\widehat \eta)\}_{k=0}^{\infty}$, where $S$ is the time shift and $f$ is strictly local in space and belongs to a class of functionals related to the H\"older continuous functions on the torus $T^{1}$.

Key words: Markov chains, stochastic operator, mixing, random walks in random environment.


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Article Information

TitleWeak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 4, 302-314
MathSciNet MR3469530
zbMATH 06630276
MilestonesReceived 17/04/2015
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

C. Boldrighini
Dipartimento di Matematica G. Castelnuovo, Sapienza Universita di Roma, Piazzale Aldo Moro 5, 00185 Roma; GNFM, Istituto Nazionale di Alta Matematica, Piazzale Aldo Moro 5, 00185 Roma

A. Marchesiello
Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Decin Branch, Pohranicni 1, 40501 Decin

C. Saffirio
Institut fur Mathematik Universit at Zurich, Winterthurerstrasse 190, CH-8057 Zurich

 


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Citation Example

C. Boldrighini, A. Marchesiello, and C. Saffirio, Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$, Methods Funct. Anal. Topology 21 (2015), no. 4, 302-314.


BibTex

@article {MFAT789,
    AUTHOR = {Boldrighini, C. and Marchesiello, A. and Saffirio, C.},
     TITLE = {Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {4},
     PAGES = {302-314},
      ISSN = {1029-3531},
  MRNUMBER = {MR3469530},
 ZBLNUMBER = {06630276},
       URL = {http://mfat.imath.kiev.ua/article/?id=789},
}


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