# deviation bounds for additive functionals of markov processes

Patrick Cattiaux; Arnaud Guillin

ESAIM: Probability and Statistics (2007)

- Volume: 12, page 12-29
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topCattiaux, Patrick, and Guillin, Arnaud. "deviation bounds for additive functionals of markov processes." ESAIM: Probability and Statistics 12 (2007): 12-29. <http://eudml.org/doc/104392>.

@article{Cattiaux2007,

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.).
},

author = {Cattiaux, Patrick, Guillin, Arnaud},

journal = {ESAIM: Probability and Statistics},

keywords = {Deviation inequalities; functional inequalities; additive functionals.; deviation inequalities; additive functionals},

language = {eng},

month = {11},

pages = {12-29},

publisher = {EDP Sciences},

title = {deviation bounds for additive functionals of markov processes},

url = {http://eudml.org/doc/104392},

volume = {12},

year = {2007},

}

TY - JOUR

AU - Cattiaux, Patrick

AU - Guillin, Arnaud

TI - deviation bounds for additive functionals of markov processes

JO - ESAIM: Probability and Statistics

DA - 2007/11//

PB - EDP Sciences

VL - 12

SP - 12

EP - 29

AB -
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.).

LA - eng

KW - Deviation inequalities; functional inequalities; additive functionals.; deviation inequalities; additive functionals

UR - http://eudml.org/doc/104392

ER -

## References

top- S. Aida, Uniform 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.
- F. Barthe, P. Cattiaux and C. Roberto, Concentration for independent random variables with heavy tails. AMRX2005 (2005) 39–60.
- F. Barthe, P. Cattiaux and C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iber.22 (2006) 993–1067.
- F. Barthe, P. Cattiaux and C. Roberto, Isoperimetry between exponential and Gaussian. EJP12 (2007) 1212–1237.
- W. Bryc and A. Dembo, Large deviations for quadratic functionals of gaussian processes. J. Theoret. Prob.10 (1997) 307–332.
- P. Cattiaux, I. Gentil and G. Guillin, Weak logarithmic-Sobolev inequalities and entropic convergence. Prob. Theory Related Fields139 (2007) 563–603.
- 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).
- H. Djellout, A. Guillin and L. Wu, Transportation 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).
- B. Franchi, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. T.A.M.S.327 (1991) 125–158.
- F.Z. Gong and F.Y. Wang, Functional inequalities for uniformly integrable semigroups and applications to essential spectrums. Forum Math.14 (2002) 293–313.
- C. Léonard, Convex conjugates of integral functionals. Acta Math. Hungar.93 (2001) 253–280.
- C. Léonard, Minimizers of energy functionals. Acta Math. Hungar.93 (2001) 281–325.
- P. Lezaud, Chernoff and Berry-Eessen inequalities for Markov processes. ESAIM Probab. Statist.5 (2001) 183–201.
- G. Lu, Weighted Poincaré 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).
- R.T. Rockafellar, Integrals which are convex functionals. Pacific J. Math.24 (1968) 525–539.
- R.T. Rockafellar, Integrals which are convex functionals II. Pacific J. Math.39 (1971) 439–469.
- M. Röckner and F.Y. Wang, Weak Poincaré inequalities and L2-convergence rates of Markov semigroups. J. Funct. Anal.185 (2001) 564–603.
- G. Royer, Une initiation aux inégalités de Sobolev logarithmiques. S.M.F., Paris (1999).
- F.Y. Wang, Functional inequalities for empty essential spectrum. J. Funct. Anal.170 (2000) 219–245.
- L. Wu, A deviation inequality for non-reversible Markov process. Ann. Inst. Henri Poincaré. Prob. Stat.36 (2000) 435–445.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.