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.
Mots clés : diffusion process, recurrence, additive functionals, ergodic theorem, polynomial convergence, hitting times, Kac formula, deviations inequalities
@article{AIHPB_2011__47_2_425_0, author = {L\"ocherbach, Eva and Loukianova, Dasha and Loukianov, Oleg}, title = {Polynomial bounds in the {Ergodic} theorem for one-dimensional diffusions and integrability of hitting times}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {425--449}, publisher = {Gauthier-Villars}, volume = {47}, number = {2}, year = {2011}, doi = {10.1214/10-AIHP359}, mrnumber = {2814417}, zbl = {1220.60045}, language = {en}, url = {http://www.numdam.org/articles/10.1214/10-AIHP359/} }
TY - JOUR AU - Löcherbach, Eva AU - Loukianova, Dasha AU - Loukianov, Oleg TI - Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2011 SP - 425 EP - 449 VL - 47 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/10-AIHP359/ DO - 10.1214/10-AIHP359 LA - en ID - AIHPB_2011__47_2_425_0 ER -
%0 Journal Article %A Löcherbach, Eva %A Loukianova, Dasha %A Loukianov, Oleg %T Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times %J Annales de l'I.H.P. Probabilités et statistiques %D 2011 %P 425-449 %V 47 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/10-AIHP359/ %R 10.1214/10-AIHP359 %G en %F AIHPB_2011__47_2_425_0
Löcherbach, Eva; Loukianova, Dasha; Loukianov, Oleg. Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 425-449. doi : 10.1214/10-AIHP359. http://www.numdam.org/articles/10.1214/10-AIHP359/
[1] A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 (2008) 1000-1034. | MR | Zbl
.[2] Passage time moments for multidimensional diffusions. J. Appl. Probab. 37 (2000) 246-251. | MR | Zbl
and .[3] Probability inequalities for sums of independent random variables. J. Amer. Statist. Assoc. 57 (1962) 33-45. | Zbl
.[4] Sharp bounds for the tails of functionals of Markov chains. Teor. Veroyatnost. i Primenen. 54 (2009) 609-619. | MR | Zbl
and .[5] Handbook of Brownian Motion: Facts and Formulae. Birkhäuser, Basel, 2002. | MR | Zbl
and .[6] Exponential moments for hitting times of uniformly ergodic Markov processes. Ann. Probab. 11 (1983) 648-665. | MR | Zbl
and .[7] Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Stat. 12 (2008) 12-29. | Numdam | MR | Zbl
and .[8] Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields 140 (2008) 19-40. | MR | Zbl
, and .[9] Concentration inequalities for Markov processes via coupling. Electron. J. Probab. 14 (2009) 1162-1180. | MR | Zbl
and .[10] 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] Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13 (2007) 514-543. | MR | Zbl
, and .[12] The first passage problem for a continuous Markov process. Ann. Math. Statist. 24 (1953) 624-639. | MR | Zbl
and .[13] Comportement des temps d'atteinte d'une diffusion fortement rentrante. Semin. Probab. 31 (1997) 168-175. | Numdam | MR | Zbl
and .[14] A result on the first-passage time of an Ornstein-Uhlenbeck process. Statist. Probab. Lett. 77 (2007) 1744-1749. | MR | Zbl
.[15] Subgeometric rates of convergence of f-ergodic strong Markov processes. Stochastic Process. Appl. 119 (2009) 897-923. | MR | Zbl
, and .[16] Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 (2004) 1353-1377. | MR | Zbl
, , and .[17] Bounds on regeneration times and limit theorems for subgeometric Markov chains. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 239-257. | Numdam | MR | Zbl
, and .[18] Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 (1995) 1671-1691. | MR | Zbl
, and .[19] 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
and .[20] Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15 (2005) 1565-1589. | MR | Zbl
and .[21] Uniform concentration inequality for ergodic diffusion process. Stochastic Process. Appl. 117 (2007) 830-839. | MR | Zbl
and .[22] 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/.
and .[23] Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift. Ann. Statist. 30 (2002) 731-753. | MR | Zbl
, and .[24] Some remarks on the Raleigh process. J. Appl. Probab. 23 (1986) 398-408. | MR | Zbl
, , and .[25] Transportation-information inequalities for Markov processes. Probab. Theory Related Fields 144 (2009) 669-695. | MR | Zbl
, , and .[26] Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Aalphen, 1980. | MR
.[27] Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13-90. | MR | Zbl
.[28] Limit Theorems for Null Recurrent Markov Processes. Mem. Amer. Math. Soc. 768. Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl
and .[29] Polynomial convergence rate of Markov chains. Ann. Appl. Probab. 12 (2002) 224-247. | MR | Zbl
and .[30] Quelques remarques sur l'ultracontractivité. J. Funct. Anal. 111 (1993) 155-196. | MR | Zbl
, and .[31] Brownian Motion and Stochastic Calculus, 2nd edition. Springer, New York, 1991. | MR | Zbl
and .[32] Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (2003) 304-362. | MR | Zbl
and .[33] Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10 (2005) 61-123. | MR | Zbl
and .[34] Chernoff and Berry-Esséen inequalities for Markov processes. ESAIM Probab. Statist. 5 (2001) 183-201. | Numdam | MR | Zbl
.[35] 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
and .[36] 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/.
, and .[37] Poincaré inequality and exponential integrability of hitting times for one-dimensional diffusion. Available at arXiv:0907.0762.
, and .[38] Some properties of one-dimensional diffusion processes. Mem. Fac. Sci. Kyusyu Univ. Ser. A Math. 11 (1957) 117-141. | MR | Zbl
and .[39] Markov Chains and Stochastic Stability. Cambridge Univ. Press, Cambridge, 2009. | MR | Zbl
and .[40] On the Poisson equation and diffusion approximation I. Ann. Probab. 29 (2001) 1061-1085. | MR | Zbl
and .[41] On the Poisson equation and diffusion approximation III. Ann. Probab. 33 (2005) 1111-1133. | MR | Zbl
and .[42] Sums of Independent Random Variables. Springer, Berlin, 1975. | MR | Zbl
.[43] On the Bennet-Hoeffding inequality. Available at arXiv:0902.4058v1[math.PR].
.[44] Continuous Martingales and Brownian Motion, 2nd edition. Springer, Berlin, 1994. | MR | Zbl
and .[45] Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl. 80 (1999) 211-229. | MR | Zbl
and .[46] Subgeometric rates of convergence off-ergodic Markov chains. Adv. in Appl. Probab. 26 (1994) 775-798. | MR | Zbl
and .[47] On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 (1997) 115-127. | MR | Zbl
.[48] On subexponential mixing rate for Markov processes. Teor. Veroyatnost. i Primenen. 49 (2004) 21-35. | MR | Zbl
and .[49] A deviation inequality for non-reversible Markov process. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 435-445. | Numdam | MR | Zbl
.Cité par Sources :