Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times

Löcherbach, Eva; Loukianova, Dasha; Loukianov, Oleg

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 425-449.

Considérons une diffusion récurrente positive avec loi initiale ν et probabilité invariante μ. Pour tout a∈ℝ, soit Ta le temps d'atteinte du point a. Supposons qu'il existe p>1 et un point a∈ℝ tels que pour tout x∈ℝ, et . Alors nous obtenons l'inégalité de déviation non-asymptotique suivante: ℙν(|(1/t)∫0tf(Xs) ds-μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p, où f est une fonction bornée ou une fonction bornée à support compact. Ici, A(f)=‖f‖∞ dans le cas d'une fonction bornée et A(f)=μ(|f|) dans le cas d'une fonction bornée à support compact. De plus, sous certaines conditions sur les coefficients de la diffusion, nous obtenons une minoration et majoration, polynomiale en x, de . Ce résultat est basé sur une formule de Kac généralisée (voir théorème 4.1) pour les moments où f est une fonction dérivable.

Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p>1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds-μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p. Here f bounded or bounded and compactly supported and A(f)=‖f‖∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under some conditions on the coefficients of X, a polynomial control of from above and below. This control is based on a generalized Kac's formula (see Theorem 4.1) for the moments of a differentiable function f.

[1] R. Adamczak. A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 (2008) 1000-1034. | MR | Zbl

[2] S. Balaji and S. Ramasubramanian. Passage time moments for multidimensional diffusions. J. Appl. Probab. 37 (2000) 246-251. | MR | Zbl

[3] G. Bennett. Probability inequalities for sums of independent random variables. J. Amer. Statist. Assoc. 57 (1962) 33-45. | Zbl

[4] P. Bertail and S. Clémençon. Sharp bounds for the tails of functionals of Markov chains. Teor. Veroyatnost. i Primenen. 54 (2009) 609-619. | MR | Zbl

[5] A. N. Borodin and P. Salminen. Handbook of Brownian Motion: Facts and Formulae. Birkhäuser, Basel, 2002. | MR | Zbl

[6] R. Carmona and A. Klein. Exponential moments for hitting times of uniformly ergodic Markov processes. Ann. Probab. 11 (1983) 648-665. | MR | Zbl

[7] P. Cattiaux and A. Guillin. Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Stat. 12 (2008) 12-29. | Numdam | MR | Zbl

[8] P. Cattiaux, A. Guillin and F. Malrieu. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields 140 (2008) 19-40. | MR | Zbl

[9] J. R. Chazottes and F. Redig. Concentration inequalities for Markov processes via coupling. Electron. J. Probab. 14 (2009) 1162-1180. | MR | Zbl

[10] S. Clémençon. Moment and probability inequalities for sums of bounded additive functionals of regular Markov chains via the Nummelin splitting technique. Statist. Probab. Lett. 55 (2001) 227-238. | MR | Zbl

[11] F. Comte, V. Genon-Catalot and Y. Rozenholc. Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13 (2007) 514-543. | MR | Zbl

[12] D. A. Darling and A. J. F. Siegert. The first passage problem for a continuous Markov process. Ann. Math. Statist. 24 (1953) 624-639. | MR | Zbl

[13] M. Deaconu and S. Wantz. Comportement des temps d'atteinte d'une diffusion fortement rentrante. Semin. Probab. 31 (1997) 168-175. | Numdam | MR | Zbl

[14] S. Ditlevsen. A result on the first-passage time of an Ornstein-Uhlenbeck process. Statist. Probab. Lett. 77 (2007) 1744-1749. | MR | Zbl

[15] R. Douc, G. Fort and A. Guillin. Subgeometric rates of convergence of f-ergodic strong Markov processes. Stochastic Process. Appl. 119 (2009) 897-923. | MR | Zbl

[16] R. Douc, G. Fort, E. Moulines and P. Soulier. Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 (2004) 1353-1377. | MR | Zbl

[17] R. Douc, A. Guillin and E. Moulines. Bounds on regeneration times and limit theorems for subgeometric Markov chains. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 239-257. | Numdam | MR | Zbl

[18] N. Down, S. P. Meyn and R. L. Tweedie. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 (1995) 1671-1691. | MR | Zbl

[19] P. J. Fitzsimmons and J. Pitman. Kac's moment formula and the Feyman-Kac formula for additive functionals of a Markov process. Stochastic Process. Appl. 79 (1999) 117-134. | MR | Zbl

[20] G. Fort and G. O. Roberts. Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15 (2005) 1565-1589. | MR | Zbl

[21] L. Galtchouk and S. Pergamenshchikov. Uniform concentration inequality for ergodic diffusion process. Stochastic Process. Appl. 117 (2007) 830-839. | MR | Zbl

[22] L. Galtchouk and S. Pergamenshchikov. Adaptive sequential estimation for ergodic diffusion processes in quadratic metric. Part 1: Sharp non-asymptotic oracle inequalities. Available at http://hal.archives-ouvertes.fr/hal-00177875/fr/.

[23] V. Genon-Catalot, C. Laredo and M. Nussbaum. Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift. Ann. Statist. 30 (2002) 731-753. | MR | Zbl

[24] V. Giorno, A. G. Nobile, L. Riccardi and L. Sacredote. Some remarks on the Raleigh process. J. Appl. Probab. 23 (1986) 398-408. | MR | Zbl

[25] A. Guillin, C. Léonard, L. Wu and N. Yao. Transportation-information inequalities for Markov processes. Probab. Theory Related Fields 144 (2009) 669-695. | MR | Zbl

[26] R. Z. Has'Minskii. Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Aalphen, 1980. | MR

[27] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13-90. | MR | Zbl

[28] R. Höpfner and E. Löcherbach. Limit Theorems for Null Recurrent Markov Processes. Mem. Amer. Math. Soc. 768. Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl

[29] S. F. Jarner and G. O. Roberts. Polynomial convergence rate of Markov chains. Ann. Appl. Probab. 12 (2002) 224-247. | MR | Zbl

[30] O. Kavian, G. Kerkyacharian and B. Roynette. Quelques remarques sur l'ultracontractivité. J. Funct. Anal. 111 (1993) 155-196. | MR | Zbl

[31] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer, New York, 1991. | MR | Zbl

[32] I. Kontoyiannis and S. P. Meyn. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (2003) 304-362. | MR | Zbl

[33] I. Kontoyiannis and S. P. Meyn. Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10 (2005) 61-123. | MR | Zbl

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

[35] E. Löcherbach and D. Loukianova. On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stochastic Process. Appl. 118 (2008) 1301-1321. | MR | Zbl

[36] E. Löcherbach, O. Loukianov and D. Loukianova. Penalized nonparametric drift estimation in a continuous time one-dimensional diffusion process. ESAIM Probab. Statist. (2010). To appear. Available at http://hal.archives-ouvertes.fr/hal-00367993/fr/.

[37] O. Loukianov, D. Loukianova and S. Song. Poincaré inequality and exponential integrability of hitting times for one-dimensional diffusion. Available at arXiv:0907.0762.

[38] G. Maruyama and H. Tanaka. Some properties of one-dimensional diffusion processes. Mem. Fac. Sci. Kyusyu Univ. Ser. A Math. 11 (1957) 117-141. | MR | Zbl

[39] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Cambridge Univ. Press, Cambridge, 2009. | MR | Zbl

[40] E. Pardoux and A. Y. Veretennikov. On the Poisson equation and diffusion approximation I. Ann. Probab. 29 (2001) 1061-1085. | MR | Zbl

[41] E. Pardoux and A. Y. Veretennikov. On the Poisson equation and diffusion approximation III. Ann. Probab. 33 (2005) 1111-1133. | MR | Zbl

[42] V. V. Petrov. Sums of Independent Random Variables. Springer, Berlin, 1975. | MR | Zbl

[43] I. Pinelis. On the Bennet-Hoeffding inequality. Available at arXiv:0902.4058v1[math.PR].

[44] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 2nd edition. Springer, Berlin, 1994. | MR | Zbl

[45] G. O. Roberts and R. L. Tweedie. Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl. 80 (1999) 211-229. | MR | Zbl

[46] P. Tuominen and R. Tweedie. Subgeometric rates of convergence off-ergodic Markov chains. Adv. in Appl. Probab. 26 (1994) 775-798. | MR | Zbl

[47] A. Y. Veretennikov. On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 (1997) 115-127. | MR | Zbl

[48] A. Y. Veretennikov and S. A. Klokov. On subexponential mixing rate for Markov processes. Teor. Veroyatnost. i Primenen. 49 (2004) 21-35. | MR | Zbl

[49] L. Wu. A deviation inequality for non-reversible Markov process. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 435-445. | Numdam | MR | Zbl