Deviation bounds for additive functionals of Markov processes
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 12-29.

In this paper we derive non asymptotic deviation bounds for

 ${¶}_{\nu }\left(|\frac{1}{t}{\int }_{0}^{t}V\left({X}_{s}\right)\mathrm{d}s-\int V\mathrm{d}\mu |\ge R\right)$
where $X$ is a $\mu$ stationary and ergodic Markov process and $V$ is some $\mu$ integrable function. These bounds are obtained under various moments assumptions for $V$, and various regularity assumptions for $\mu$. Regularity means here that $\mu$ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc.).

DOI : https://doi.org/10.1051/ps:2007032
Classification : 60F10,  60J25
Mots clés : deviation inequalities, functional inequalities, additive functionals
@article{PS_2008__12__12_0,
author = {Cattiaux, Patrick and Guillin, Arnaud},
title = {Deviation bounds for additive functionals of {Markov} processes},
journal = {ESAIM: Probability and Statistics},
pages = {12--29},
publisher = {EDP-Sciences},
volume = {12},
year = {2008},
doi = {10.1051/ps:2007032},
mrnumber = {2367991},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2007032/}
}
TY  - JOUR
AU  - Cattiaux, Patrick
AU  - Guillin, Arnaud
TI  - Deviation bounds for additive functionals of Markov processes
JO  - ESAIM: Probability and Statistics
PY  - 2008
DA  - 2008///
SP  - 12
EP  - 29
VL  - 12
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007032/
UR  - https://www.ams.org/mathscinet-getitem?mr=2367991
UR  - https://doi.org/10.1051/ps:2007032
DO  - 10.1051/ps:2007032
LA  - en
ID  - PS_2008__12__12_0
ER  - 
Cattiaux, Patrick; Guillin, Arnaud. Deviation bounds for additive functionals of Markov processes. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 12-29. doi : 10.1051/ps:2007032. http://www.numdam.org/articles/10.1051/ps:2007032/

[1] S. Aida, Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal. 158 (1998) 152-185. | MR 1641566 | Zbl 0914.47041

[2] 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. | MR 1307413 | Zbl 0856.47026

[3] F. Barthe, P. Cattiaux and C. Roberto, Concentration for independent random variables with heavy tails. AMRX 2005 (2005) 39-60. | MR 2173316 | Zbl 1094.60010

[4] F. Barthe, P. Cattiaux and C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iber. 22 (2006) 993-1067. | MR 2320410 | Zbl 1118.26014

[5] F. Barthe, P. Cattiaux and C. Roberto, Isoperimetry between exponential and Gaussian. EJP 12 (2007) 1212-1237. | MR 2346509 | Zbl 1132.26005

[6] W. Bryc and A. Dembo, Large deviations for quadratic functionals of gaussian processes. J. Theoret. Prob. 10 (1997) 307-332. | MR 1455147 | Zbl 0894.60026

[7] P. Cattiaux, I. Gentil and G. Guillin, Weak logarithmic-Sobolev inequalities and entropic convergence. Prob. Theory Related Fields 139 (2007) 563-603. | MR 2322708 | Zbl 1130.26010

[8] E.B. Davies, Heat kernels and spectral theory. Cambridge University Press (1989). | MR 990239 | Zbl 0699.35006

[9] J.D. Deuschel and D.W. Stroock, Large Deviations. Academic Press, London, Pure Appl. Math. 137 (1989). | MR 997938 | Zbl 0705.60029

[10] H. Djellout, A. Guillin and L. Wu, Transportation cost information inequalities for random dynamical systems and diffusions. Ann. Prob. 334 (2002) 1025-1028. | Zbl 1061.60011

[11] P. Doukhan, Mixing, Properties and Examples. Springer-Verlag, Lect. Notes Statist. 85 (1994). | MR 1312160 | Zbl 0801.60027

[12] B. Franchi, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. T.A.M.S. 327 (1991) 125-158. | MR 1040042 | Zbl 0751.46023

[13] F.Z. Gong and F.Y. Wang, Functional inequalities for uniformly integrable semigroups and applications to essential spectrums. Forum Math. 14 (2002) 293-313. | MR 1880915

[14] C. Léonard, Convex conjugates of integral functionals. Acta Math. Hungar. 93 (2001) 253-280. | MR 1925355 | Zbl 0997.52008

[15] C. Léonard, Minimizers of energy functionals. Acta Math. Hungar. 93 (2001) 281-325. | MR 1925356 | Zbl 1002.49017

[16] P. Lezaud, Chernoff and Berry-Eessen inequalities for Markov processes. ESAIM Probab. Statist. 5 (2001) 183-201. | Numdam | MR 1875670 | Zbl 0998.60075

[17] G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications. Rev. Mat. Iber. 8 (1992) 367-439. | MR 1202416 | Zbl 0804.35015

[18] E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants. Springer-Verlag, Math. Appl. 31 (2000). | MR 2117923 | Zbl 0944.60008

[19] R.T. Rockafellar, Integrals which are convex functionals. Pacific J. Math. 24 (1968) 525-539. | MR 236689 | Zbl 0159.43804

[20] R.T. Rockafellar, Integrals which are convex functionals II. Pacific J. Math. 39 (1971) 439-469. | MR 310612 | Zbl 0236.46031

[21] M. Röckner and F.Y. Wang, Weak Poincaré inequalities and ${L}^{2}$-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001) 564-603. | MR 1856277 | Zbl 1009.47028

[22] G. Royer, Une initiation aux inégalités de Sobolev logarithmiques. S.M.F., Paris (1999). | MR 1704288 | Zbl 0927.60006

[23] F.Y. Wang, Functional inequalities for empty essential spectrum. J. Funct. Anal. 170 (2000) 219-245. | MR 1736202 | Zbl 0946.58010

[24] L. Wu, A deviation inequality for non-reversible Markov process. Ann. Inst. Henri Poincaré. Prob. Stat. 36 (2000) 435-445. | Numdam | MR 1785390 | Zbl 0972.60003

Cité par Sources :