Local limit theorems for transition densities of markov chains converging to diffusions

Probability Theory and Related Fields

Volumn 117, Issue 4, 2000, Pages 551-587

  Konakov, Valentin ^a   Mammen, Enno ^b  

^a CENTRAL ECONOMICS AND MATHEMATICS INSTITUTE   (Russian Federation)

^b UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

Diffusion processes; Markov chains; Transition densities

Indexed keywords

EID: 0034389016     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00008735     Document Type: Article

References (27)

1. Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Probab. Theory Rel. Fields 104, 43-60 (1996a)
2. Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations: II. Convergence rate of the density. Monte Carlo Methods Appl. 2, 93-128 (1996b)
3. Bhattacharya, R., Rao, R.: Normal approximations and asymptotic expansions. John Wiley & Sons, New York (1976)
4. Dahlhaus, R.: Fitting time series models to nonstationary processes. Ann. Statist. 25, 1-37 (1997)
5. Davydov, Y. A.: On strong con vergence of distributions of functionals of random processes. I. Theory Probab. Appl. 25, 772-789 (1980)
6. Davydov, Y.A.: On strong convergence of distributions of functionals of random processes. II. Theory Probab. Appl. 26, 258-278 (1981)
7. Franke J., Kreiss J.-P., Mammen, E.: Bootstrap of kernel smoothing in nonlinear time series. Manuscript. (1996)
8. Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, New Jersey (1964)
9. Gel'fand, I.M., Shilov, G.E.: Generalized functions, vol.2. Fizmatgiz, Moscow (in russian) (1958)
10. Kasymdzganova, M.: Local limit theorems for homogenieus Markov chains. Ph.D. Thesis. Moscow Statw Univ. (in russian) (1981 )
11. Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Springer, Berlin, Heidelberg (1992)
12. Konakov, V.D., Mammen, E.: Local approximations of Markov random walks by diffusions. Manuscript (1999)
13. Konakov, V.D., Molchanov, S.A.: On the convergence of Markov chains to diffusion processes. Teor. Veroyatn. Mat. Stat.31, 51-64, (in russian) (English translation in Theory Probab. Math. Stat. (1985) 31, 59-73] (1984)
14. Konakov, V.D., Molchanov, S.A.: On the convergence of Markov chains to diffusion processes. Teor. Veroyatn. Mat. Stat.31, 51-64, (in russian) (English translation in Theory Probab. Math. Stat. (1985) 31, 59-73] (1984)
15. Konovalov, A.G.: A local limit theorem on convergence to a diffusion process. Theory Probab. Appl., 26, 581-587 (1981 )
16. Kutoyants, Y.A.: Efficiency of the empirical distribution for ergodic diffusion. Bernoulli 3, 445-456 (1997a)
17. Kutoyants, Y.A.: Some problems of nonparametric estimation by observations of ergodic diffusion process. Statist. Probab. Letters 32, 311-320 (1997b)
18. Kuznetsov, S.E.: Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs. Comm. Pure Appl. Math. 51, 303-340 (1998)
19. Ladyženskaja, O.A., Solonnikov, V.A. and and Ural'ceva: Linear and quasi-linear equations of parabolic type. Amer. Math. Soc., Providence, Rhode Island (1968)
20. McKean, H.P., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Differential Geometry 1,43-69(1967)
21. Pinsky, M. A. Lectures on random evolution. World Scientific, Singapore, New Jersey ( 1991 )
22. Platen, E.: An introduction to numerical methods for stochastic differential equations. Re-search Paper 6, UTS, Sidney (1999)
23. Robinson, P.M.: Nonparametric estimators for time series. J. Time Ser. Anal. 4, 185-207 (1983)
24. Stroock, D.W., Varadhan, S. R.: Multidimensional diffusion processes. Springer, Berlin, Heidelberg, New York (1979)
25. Skorohod, A.V.: Studies in the theory of random processes. Addison-Wesley, Reading, Massachussetts.[English translation of Skorohod, A. V. (1961). Issledovaniya po teorii sluchaynykh protsessov. Kiev University Press] (1965)
26. Skorohod, A. V. Asymptotic methods for Stochastic differential equations, (in russian). Kiev Naukova dumka (1987)
27. Tjøstheim, D. Non-linear time series: a selective review. Scand. J. Statist. 21, 97-130 (1994)o