Concentration of measure on product spaces with applications to Markov processes

Gordon Blower¹ ; François Bolley²

¹ Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF, UK
² École 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

Studia Mathematica, Tome 175 (2006) no. 1, pp. 47-72

Résumé

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.

Détail

DOI : 10.4064/sm175-1-3

Keywords: stochastic process state space polish space paper gives sufficient conditions initial conditional distributions joint law satisfy gaussian concentration inequalities transportation inequalities the euclidean space mathbb there sufficient conditions joint law satisfy logarithmic sobolev inequality several cases constants obtained optimal order growth respect number random variables independent number these results extend results known mutually independent random variables weakly dependent random variables under dobrushin shlosman type conditions paper contains applications markov processes including arma process