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

Eva Löcherbach; Dasha Loukianova; Oleg Loukianov

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 425-449
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Löcherbach, Eva, Loukianova, Dasha, and 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 47.2 (2011): 425-449. <http://eudml.org/doc/242012>.

@article{Löcherbach2011,
abstract = {Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p&gt;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.},
author = {Löcherbach, Eva, Loukianova, Dasha, Loukianov, Oleg},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {diffusion process; recurrence; additive functionals; ergodic theorem; polynomial convergence; hitting times; Kac formula; deviations inequalities},
language = {eng},
number = {2},
pages = {425-449},
publisher = {Gauthier-Villars},
title = {Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times},
url = {http://eudml.org/doc/242012},
volume = {47},
year = {2011},
}

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
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 425
EP - 449
AB - Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p&gt;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.
LA - eng
KW - diffusion process; recurrence; additive functionals; ergodic theorem; polynomial convergence; hitting times; Kac formula; deviations inequalities
UR - http://eudml.org/doc/242012
ER -

References

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  1. [1] R. Adamczak. A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 (2008) 1000–1034. Zbl1190.60010MR2424985
  2. [2] S. Balaji and S. Ramasubramanian. Passage time moments for multidimensional diffusions. J. Appl. Probab. 37 (2000) 246–251. Zbl0965.60068MR1761674
  3. [3] G. Bennett. Probability inequalities for sums of independent random variables. J. Amer. Statist. Assoc. 57 (1962) 33–45. Zbl0104.11905
  4. [4] P. Bertail and S. Clémençon. Sharp bounds for the tails of functionals of Markov chains. Teor. Veroyatnost. i Primenen. 54 (2009) 609–619. Zbl1211.60028MR2766354
  5. [5] A. N. Borodin and P. Salminen. Handbook of Brownian Motion: Facts and Formulae. Birkhäuser, Basel, 2002. Zbl1012.60003MR1912205
  6. [6] R. Carmona and A. Klein. Exponential moments for hitting times of uniformly ergodic Markov processes. Ann. Probab. 11 (1983) 648–665. Zbl0523.60064MR704551
  7. [7] P. Cattiaux and A. Guillin. Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Stat. 12 (2008) 12–29. Zbl1183.60011MR2367991
  8. [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. Zbl1169.35031MR2357669
  9. [9] J. R. Chazottes and F. Redig. Concentration inequalities for Markov processes via coupling. Electron. J. Probab. 14 (2009) 1162–1180. Zbl1191.60023MR2511280
  10. [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. Zbl1078.60508MR1867526
  11. [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. Zbl1127.62067MR2331262
  12. [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. Zbl0053.27301MR58908
  13. [13] M. Deaconu and S. Wantz. Comportement des temps d’atteinte d’une diffusion fortement rentrante. Semin. Probab. 31 (1997) 168–175. Zbl0882.60077MR1478725
  14. [14] S. Ditlevsen. A result on the first-passage time of an Ornstein–Uhlenbeck process. Statist. Probab. Lett. 77 (2007) 1744–1749. Zbl1133.60314MR2394571
  15. [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. Zbl1163.60034MR2499863
  16. [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. Zbl1082.60062MR2071426
  17. [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. Zbl1176.60063MR2446322
  18. [18] N. Down, S. P. Meyn and R. L. Tweedie. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 (1995) 1671–1691. Zbl0852.60075MR1379163
  19. [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. Zbl0962.60067MR1670526
  20. [20] G. Fort and G. O. Roberts. Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15 (2005) 1565–1589. Zbl1072.60057MR2134115
  21. [21] L. Galtchouk and S. Pergamenshchikov. Uniform concentration inequality for ergodic diffusion process. Stochastic Process. Appl. 117 (2007) 830–839. Zbl1117.60026MR2330721
  22. [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/. Zbl05930607
  23. [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. Zbl1029.62071MR1922540
  24. [24] V. Giorno, A. G. Nobile, L. Riccardi and L. Sacredote. Some remarks on the Raleigh process. J. Appl. Probab. 23 (1986) 398–408. Zbl0598.60085MR839994
  25. [25] A. Guillin, C. Léonard, L. Wu and N. Yao. Transportation-information inequalities for Markov processes. Probab. Theory Related Fields 144 (2009) 669–695. Zbl1169.60304MR2496446
  26. [26] R. Z. Has’minskii. Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Aalphen, 1980. MR600653
  27. [27] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13–90. Zbl0127.10602MR144363
  28. [28] R. Höpfner and E. Löcherbach. Limit Theorems for Null Recurrent Markov Processes. Mem. Amer. Math. Soc. 768. Amer. Math. Soc., Providence, RI, 2003. Zbl1018.60074MR1949295
  29. [29] S. F. Jarner and G. O. Roberts. Polynomial convergence rate of Markov chains. Ann. Appl. Probab. 12 (2002) 224–247. Zbl1012.60062MR1890063
  30. [30] O. Kavian, G. Kerkyacharian and B. Roynette. Quelques remarques sur l’ultracontractivité. J. Funct. Anal. 111 (1993) 155–196. Zbl0807.47027MR1200640
  31. [31] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer, New York, 1991. Zbl0638.60065MR1121940
  32. [32] I. Kontoyiannis and S. P. Meyn. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (2003) 304–362. Zbl1016.60066MR1952001
  33. [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. Zbl1079.60067MR2120240
  34. [34] P. Lezaud. Chernoff and Berry–Esséen inequalities for Markov processes. ESAIM Probab. Statist. 5 (2001) 183–201. Zbl0998.60075MR1875670
  35. [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. Zbl1202.60122MR2427041
  36. [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/. Zbl06157514
  37. [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. Zbl1233.60044
  38. [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. Zbl0089.34604MR97128
  39. [39] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Cambridge Univ. Press, Cambridge, 2009. Zbl1165.60001MR2509253
  40. [40] E. Pardoux and A. Y. Veretennikov. On the Poisson equation and diffusion approximation I. Ann. Probab. 29 (2001) 1061–1085. Zbl1029.60053MR1872736
  41. [41] E. Pardoux and A. Y. Veretennikov. On the Poisson equation and diffusion approximation III. Ann. Probab. 33 (2005) 1111–1133. Zbl1071.60022MR2135314
  42. [42] V. V. Petrov. Sums of Independent Random Variables. Springer, Berlin, 1975. Zbl0322.60042MR388499
  43. [43] I. Pinelis. On the Bennet–Hoeffding inequality. Available at arXiv:0902.4058v1[math.PR]. 
  44. [44] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 2nd edition. Springer, Berlin, 1994. Zbl0917.60006MR1303781
  45. [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. Zbl0961.60066MR1682243
  46. [46] P. Tuominen and R. Tweedie. Subgeometric rates of convergence off-ergodic Markov chains. Adv. in Appl. Probab. 26 (1994) 775–798. Zbl0803.60061MR1285459
  47. [47] A. Y. Veretennikov. On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 (1997) 115–127. Zbl0911.60042MR1472961
  48. [48] A. Y. Veretennikov and S. A. Klokov. On subexponential mixing rate for Markov processes. Teor. Veroyatnost. i Primenen. 49 (2004) 21–35. Zbl1090.60067MR2141328
  49. [49] L. Wu. A deviation inequality for non-reversible Markov process. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 435–445. Zbl0972.60003MR1785390

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