On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

Stochastic Processes and their Applications

Volumn 118, Issue 8, 2008, Pages 1301-1321

  Löcherbach, Eva ^a   Loukianova, Dasha ^b  

^a UNIVERSITÉ PARIS EST CRÉTEIL   (France)

^b UNIVERSITÉ D EVRY VAL D ESSONNE   (France)

Author keywords

Additive functionals; Chacon Ornstein theorem; Continuous time Markov processes; Diffusion process; Harris recurrence; Nadaraya Watson estimator; Nummelin splitting; Resolvents; Special functions

Indexed keywords

CONTINUOUS TIME SYSTEMS; ESTIMATION; FOOD ADDITIVES; LIFE CYCLE; PARAMETER ESTIMATION; SEMICONDUCTOR DOPING;

(E ,3E) PROCESS; (T ,S)-SPLITTING; CONTINUOUS-TIME (CT); DETERMINISTIC EQUIVALENTS; DIFFUSION PROCESSES; ELSEVIER (CO); ERGODIC; FUNCTIONALS; KERNEL ESTIMATION; LIFE CYCLE (LC); MULTI DIMENSIONAL; NON-PARAMETRIC; SPLITTING TECHNIQUES;

MARKOV PROCESSES;

EID: 46049086561     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2007.09.003     Document Type: Article

References (25)

1. Athreya K.B., and Ney P. A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 (1978) 493-501
2. Bhattacharya R.N. Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab. 6 4 (1978) 541-553
3. Brancovan M. Fonctionnelles additives spéciales des processus récurrents au sens de Harris. Z. Wahrscheinlichkeitstheor. Verw. Geb. 47 (1979) 163-194
4. Chen X. How often does a Harris recurrent Markov chain recur?. Ann. Probab. 27 (1999) 1324-1346
5. Chen X. The law of the iterated logarithm for functionals of Harris recurrent Markov chains: Self normalization. J. of Theoretical Probab. 12 2 (1999) 421-445
6. Csáki E., and Salminen P. On the additive functionals of diffusion processes. Studia Math. Hungar. 31 (1996) 47-62
7. Dalalyan A., and Reiss M. Asymptotic equivalence for ergodic diffusions: the multi-dimensional case. Probab. Theory Rel. Fields 134 2 (2006) 248-282
8. S. Delattre, M. Hoffmann, M. Kessler, Dynamics adaptive estimation of a scalar diffusion. Preprint 762 of Laboratoire de Probabilités et modèles aléatoires, October 2002
9. Has'minskii R.Z. Stochastic Stability of Differential Equations (1980), Sijthoff and Noordhoff, Aalphen
10. Höpfner R., and Kutoyants Y.A. On a problem of statistical inference in null recurrent diffusions. Stat. Inference Stoch. Process. 6 1 (2003) 25-42
11. Höpfner R., and Löcherbach E. Limit theorems for null recurrent Markov processes. Mem. Amer. Math. Soc. 161 768 (2003)
12. Ikeda N., Nagasawa M., and Watanabe S. A construction of Markov processes by piecing out. Proc. Japan Acad. 42 (1966) 370-375
13. Khasminskii R.Z. Asymptotic behaviour of parabolic equations arising from one-dimensional null recurrent diffusions. J. Differential Equations 161 (2000) 154-173
14. Khasminskii R.Z. Limit distributions for some integral functionals for null recurrent diffusions. Stochastic Process Appl. 92 (2001) 1-9
15. Kusuoka S., and Stroock D. Applications of the Malliavin calculus, Part II. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 32 (1985) 1-76
16. Kutoyants Y.A. Efficient density estimation for ergodic diffusion processes. Stat. Inference Stoch. Process. 1 2 (1998) 131-155
17. Kutoyants Y.A. Statistical Inference for Ergodic Diffusion Processes. Springer Series in Statistics (2004), Springer, London
18. D. Loukianova, O. Loukianov, Deterministic equivalent of additive functionals of recurrent diffusions and drift estimation. Stat. Inference Stoch. Process (2008) (in press)
19. Meyn S.P., and Tweedie R.L. Generalized resolvents and Harris recurrence of Markov processes. In: Cohn H. (Ed). Doeblin and modern probability. Proceedings of the Doeblin conference '50 years after Doeblin: development in the theory of Markov chains, Markov processes, and sums of random variables'. Contemp. Math. vol. 149 (1993), American Mathematical Society, Providence, RI 227-250
20. S.P. Meyn, R.L. Tweedie, A survey of Foster-Lyapunov techniques for general state space Markov processes, in: Proceedings of the Workshop on Stochastic Stability and Stochastic Stabilization, Metz, France, June 1993
21. Nummelin E. A splitting technique for Harris recurrent Markov chains. Z. Wahrscheinlichkeitstheor. Verw. Geb. 43 (1978) 309-318
22. Orey S. Limit Theorems for Markov Chain Transition Probabilities (1971), Van Nostrand Reinhold, London
23. Revuz D. Markov Chains. Revised ed. (1984), North Holland, Amsterdam
24. Touati A. Théorèmes limites pour les processus de Markov récurrents, Unpublished paper 1988; See also. C. R. Acad. Sci. Paris Série I 305 (1987) 841-844
25. Williams D. To begin at the beginning .... Stochastic integrals. (Proc. Durham 1980). Lecture Notes in Math. vol. 851 (1981), Springer 1-55