Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions

Loukianova, D.; Loukianov, O.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 771-786.

Habituellement le problème de l'estimation du drift pour un processus de diffusion est considéré sous l'hypothèse de l'ergodicité. Il l'est moins souvent sous l'hypothèse de nulle-récurrence, car dans ce cas il y a moins de théorèmes limites, et ceux qui existent ne s'appliquent pas à toute la classe nulle-récurrente. Le but de cet article est de démontrer quelques théorèmes limites pour les fonctionnelles additives et martingales dépendantes d'une diffusion récurrente générale (ergodique ou nulle). Ces théorèmes permettent de donner une approche unifiée au problème de l'estimation non-paramétrique par noyau du drift dans le cas unidimensionnel récurrent. Comme exemple on obtient la vitesse de convergence de l'estimateur de Nadaraya-Watson dans le cas d'un drift localement hölderien.

Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya-Watson estimator in the case of a locally Hölder-continuous drift.

Classification : 60G17, 60F10, 92B20, 68T10
Mots clés : Harris recurrence, diffusion processes, limit theorems, additive functionals, non-parametric estimation, Nadaraya-Watson estimator, rate of convergence

[1] A. Borodin and P. Salminen. Handbook of Brownian Motion - Facts and Formulae. Probability and its Applications. Birkhäuser, Basel, 1996. | MR | Zbl

[2] M. Brancovan. Fonctionnelles additives spéciales des processus récurrents au sens de Harris. Z. Wahrsch. Verw. Gebiete 47 (1979) 163-194. | MR | Zbl

[3] X. Chen. How often does a Harris recurrent Markov chain recur? Ann. Probab. 27 (1999) 1324-1346. | MR | Zbl

[4] A. Dalalyan. Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Statist. 33 (2005) 2507-2528. | MR | Zbl

[5] A. Dalalyan and Y. Kutoyants. On second order minimax estimation of the invariant density for ergodic diffusions. Statist. Decisions 22 (2004) 17-41. | MR | Zbl

[6] S. Delattre and M. Hoffmann. Asymptotic equivalence for a null recurrent diffusion model. Bernoulli 8 (2002) 139-174. | MR | Zbl

[7] S. Delattre, M. Hoffmann and M. Kessler. Dynamics adaptive estimation of a scalar diffusion. Prépublication PMA-762, Univ. Paris 6. Available at www.proba.jussieu.fr/mathdoc/preprints/.

[8] L. Galtchouk and S. Pergamentchikov. Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes. Math. Methods Statist. 10 (2001) 316-330. | MR | Zbl

[9] R. Höpfner and Y. Kutoyants. On a problem of statistical inference in null recurrent diffusions. Stat. Inference Stoch. Process. 6 (2003) 25-42. | MR | Zbl

[10] R. Höpfner and E. Löcherbach. Limit Theorems for Null Recurrent Markov Processes. Providence, RI, 2003. | MR | Zbl

[11] K. Itô and H. P. Mckean, Jr. Diffusion Processes and Their Sample Paths. Springer, Berlin, 1974. | MR | Zbl

[12] R. Khasminskii. Limit distributions of some integral functionals for null-recurrent diffusions. Stochastic Process. Appl. 92 (2001) 1-9. | MR | Zbl

[13] K. Kuratowski. Introduction a la theorie des ensembles et a la topologie. Institut de Mathematiques, Universite Geneve, 1966. | MR | Zbl

[14] Y. Kutoyants. Statistical Inference for Ergodic Diffusion Processes. Springer, London, 2004. | MR | Zbl

[15] E. Löcherbach and D. Loukianova. On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multidimensional diffusions. To appear in Stochastic Process. Appl.

[16] D. Loukianova and O. Loukianov. Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation. To appear in Stat. Inference Stoch. Process.

[17] D. Loukianova and O. Loukianov. Almost sure rate of convergence of maximum likelihood estimators for multidimensional diffusions. In Dependence in Probability and Statistics 269-347. Springer, New York, 2006. | MR | Zbl

[18] Y. Nishiyama. A maximum inequality for continuous martingales and M-estimation in Gaussian white noise model. Ann. Statist. 27 (1999) 675-696. | MR | Zbl

[19] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1994. | MR | Zbl

[20] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Vol. 2, Wiley, New York, 1990. | MR | Zbl

[21] A. Touati. Théorèmes limites pour les processus de Markov récurrents. Unpublished paper, 1988. (See also C.R.A.S. Paris Série I 305 (1987) 841-844.) | MR | Zbl

[22] H. Van Zanten. On empirical processes for ergodic diffusions and rates of convergence of M-estimators. Scand. J. Statist. 30 (2003) 443-458. | MR | Zbl

[23] H. Van Zanten. On the rate of convergence of the maximum likelihood estimator in Brownian semimartingale models. Bernoulli 11 (2005) 643-664. | MR | Zbl

[24] N. Yoshida. Asymptotic behavior of M-estimators and related random field for diffusion process. Ann. Inst. Statist. Math. 42 (1990) 221-251. | MR | Zbl