Deviation bounds for additive functionals of Markov processes

Cattiaux, Patrick; Guillin, Arnaud

ESAIM: Probability and Statistics, Tome 12 (2008), pp. 12-29.

In this paper we derive non asymptotic deviation bounds for

$${\P }_{\nu }\left(\right|\frac{1}{t}{\int }_0^{t}V\left(X_s\right)\mathrm{d}s-\int V\mathrm{d}\mu |\ge R)$$

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.).

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

[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

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

[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 | Zbl

[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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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