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deviation bounds for additive functionals of markov processes

Published online by Cambridge University Press:  13 November 2007

Patrick Cattiaux  and
Arnaud Guillin
Patrick Cattiaux
Affiliation:
École Polytechnique, CMAP, 91128 Palaiseau cedex, France, CNRS 756, and Université Paris X Nanterre, équipe MODAL'X, UFR SEGMI, 200 avenue de la République, 92001 Nanterre cedex, France; [email protected]
Arnaud Guillin
Affiliation:
Ceremade, Université Paris IX Dauphine, 75775 Paris cedex, France, CNRS 7534; [email protected]
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Abstract

In this paper we derive non asymptotic deviation bounds for $${\mathbb P}_\nu (|\frac 1t\int_0^t V(X_s) {\rm d}s - \int V {\rm d} \mu | \geq R)$$ where X is a μ stationary and ergodic Markov process and V is some μ integrable function. These bounds are obtained under various moments assumptions for V, and various regularity assumptions for μ. Regularity means here that μ may satisfy various functional inequalities (F-Sobolev,generalized Poincaré etc.).

Keywords

Deviation inequalitiesfunctional inequalitiesadditive functionals.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 12 - 29
Copyright
© EDP Sciences, SMAI, 2008

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References

Uniform, S. Aida positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal. 158 (1998) 152–185.
D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on Probability theory. École d'été de Probabilités de St-Flour 1992, Lect. Notes Math. 1581 (1994) 1–114.
Barthe, F., Cattiaux, P. and Roberto, C., Concentration for independent random variables with heavy tails. AMRX 2005 (2005) 39–60.
Barthe, F., Cattiaux, P. and Roberto, C., Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iber. 22 (2006) 993–1067. CrossRef
Barthe, F., Cattiaux, P. and Roberto, C., Isoperimetry between exponential and Gaussian. EJP 12 (2007) 1212–1237.
Bryc, W. and Dembo, A., Large deviations for quadratic functionals of gaussian processes. J. Theoret. Prob. 10 (1997) 307–332. CrossRef
Cattiaux, P., Gentil, I. and Guillin, G., Weak logarithmic-Sobolev inequalities and entropic convergence. Prob. Theory Related Fields 139 (2007) 563–603. CrossRef
E.B. Davies, Heat kernels and spectral theory. Cambridge University Press (1989).
J.D. Deuschel and D.W. Stroock, Large Deviations. Academic Press, London, Pure Appl. Math. 137 (1989).
Djellout, H., Guillin, A. and Transportation, L. Wu cost information inequalities for random dynamical systems and diffusions. Ann. Prob. 334 (2002) 1025–1028.
P. Doukhan, Mixing, Properties and Examples. Springer-Verlag, Lect. Notes Statist. 85 (1994).
Franchi, B., Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. T.A.M.S. 327 (1991) 125–158.
Gong, F.Z. and Wang, F.Y., Functional inequalities for uniformly integrable semigroups and applications to essential spectrums. Forum Math. 14 (2002) 293–313. CrossRef
Léonard, C., Convex conjugates of integral functionals. Acta Math. Hungar. 93 (2001) 253–280. CrossRef
Léonard, C., Minimizers of energy functionals. Acta Math. Hungar. 93 (2001) 281–325. CrossRef
Lezaud, P., Chernoff and Berry-Eessen inequalities for Markov processes. ESAIM Probab. Statist. 5 (2001) 183–201. CrossRef
Weighted Poincaré, G. Lu and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications. Rev. Mat. Iber. 8 (1992) 367–439.
E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants. Springer-Verlag, Math. Appl. 31 (2000).
Rockafellar, R.T., Integrals which are convex functionals. Pacific J. Math. 24 (1968) 525–539. CrossRef
Rockafellar, R.T., Integrals which are convex functionals II. Pacific J. Math. 39 (1971) 439–469. CrossRef
Röckner, M. and Wang, F.Y., Weak Poincaré inequalities and L 2-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001) 564–603. CrossRef
G. Royer, Une initiation aux inégalités de Sobolev logarithmiques. S.M.F., Paris (1999).
Wang, F.Y., Functional inequalities for empty essential spectrum. J. Funct. Anal. 170 (2000) 219–245. CrossRef
Wu, L., A deviation inequality for non-reversible Markov process. Ann. Inst. Henri Poincaré. Prob. Stat. 36 (2000) 435–445. CrossRef