A+ CATEGORY SCIENTIFIC UNIT

Concentration of measure on product spaces with applications to Markov processes

Volume 175 / 2006

Gordon Blower, François Bolley Studia Mathematica 175 (2006), 47-72 MSC: Primary 60E15; Secondary 60E05, 39B62. DOI: 10.4064/sm175-1-3

Abstract

For a stochastic process with state space some Polish space, this paper gives sufficient conditions on the initial and conditional distributions for the joint law to satisfy Gaussian concentration inequalities and transportation inequalities. In the case of the Euclidean space ${{{\mathbb R}}}^{m}$, there are sufficient conditions for the joint law to satisfy a logarithmic Sobolev inequality. In several cases, the constants obtained are of optimal order of growth with respect to the number of random variables, or are independent of this number. These results extend results known for mutually independent random variables to weakly dependent random variables under Dobrushin–Shlosman type conditions. The paper also contains applications to Markov processes including the ARMA process.

Authors

  • Gordon BlowerDepartment of Mathematics and Statistics
    Lancaster University
    Lancaster LA1 4YF, UK
    e-mail
  • François BolleyÉcole Normale Supérieure de Lyon
    Umpa, 46 allée d'Italie
    F-69364 Lyon Cedex 07, France
    and
    Institut de mathématiques – LSP
    Université Paul Sabatier
    F-31062 Toulouse Cedex 9, France
    e-mail

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