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Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains
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  • Published: 06 December 2007

Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains

  • Valentin Konakov1 &
  • Enno Mammen2 

Probability Theory and Related Fields volume 143, pages 137–176 (2009)Cite this article

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Abstract

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points.

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References

  1. Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Probab. Theory Relat. Fields 104, 43–60 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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 (1996)

    MATH  MathSciNet  Google Scholar 

  3. Bertail, P., Clémenčon, S.: Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains. Probab. Theory Relat. Fields 130, 388–414 (2004)

    Article  MATH  Google Scholar 

  4. Bertail, P., Clémenčon, S.: Regenerative block bootstrap for Markov chains. Bernoulli 12, 689–712 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bhattacharya, R., Rao, R.: Normal Approximations and Asymptotic Expansions. Wiley, New York (1976)

    Google Scholar 

  6. Bolthausen, E.: The Berry–Esseen theorem for functionals of discrete Markov chains. Z. Wahrsch. verw. Geb. 54, 59–73 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bolthausen, E.: The Berry–Esseen theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. verw. Geb. 60, 283–289 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  9. Fukasawa, M.: Edgeworth expansion for ergodic diffusions. Probab. Theory Relat. Fields (in print) (2007a)

  10. Fukasawa, M.: Regenerative block bootstrap for ergodic diffusions (preprint) (2007b)

  11. Götze, F.: Edgeworth expansions in functional limit theorems. Ann. Probab. 17(4), 1602–1634 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Götze, F., Hipp, C.: Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrsch. verw. Geb. 64, 211–239 (1983)

    Article  MATH  Google Scholar 

  13. Guyon, J.: Euler scheme and tempered distributions. Stoch. Proc. Appl. 116, 877–904 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jacod, J.: The Euler scheme for Levy driven stochastic differential equations: limit theorems. Ann. Probab. 32, 1830–1872 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jacod, J., Protter, P.: Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26, 267–307 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jacod, J., Kurtz, T., Meleard, S., Protter, P.: The approximate Euler method for Levy driven stochastic differential equations. Ann. de l’I.H.P. 41, 523–558 (2005)

    MATH  MathSciNet  Google Scholar 

  17. Jensen, J.L.: Asymptotic expansions for strongly mixing Harris recurrent Markov chains. Scand. J. Statist. 16, 47–63 (1989)

    MATH  MathSciNet  Google Scholar 

  18. Konakov, V., Molchanov, S.: On the convergence of Markov chains to diffusion processes. Teoria veroyatnostei i matematiceskaya statistika, 31, 51–64 (1984) (in russian) [English translatiion in Theory Probab. Math. Stat. 31, 59–73 (1985)]

  19. Konakov, V., Mammen, E.: Local limit theorems for transition densities of Markov chains converging to diffusions. Probab. Theory Relat. Fields. 117, 551–587 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Konakov, V., Mammen, E.: Edgeworth type expansions for Euler schemes for stochastic differential equations. Monte Carlo Methods Appl. 8, 271–286 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Konakov, V., Mammen, E.: Edgeworth-type expansions for transition densities of Markov chains converging to diffusions. Bernoulli 11(4), 591–641 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Konakov, V., Mammen, E.: Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains. Extended version (preprint). ArXiv: 0705.3139 (available under http://fr.arxiv.org/abs/0705.3139) (2007)

  23. Kusuoka, S., Yoshida, N.: Malliavin calculus, geometric mixing, and expansion of diffusion functionals. Probab. Theory Relat. Fields 116, 457–484 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ladyzenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.: Linear and quasi-linear equations of parabolic type. Amer. Math. Soc., Providence, Rhode Island (1968)

  25. Malinovskii, V.K.: Limit theorems for Harris Markov chains, 1. Theory Probab. Appl. 31, 269–285 (1987)

    Article  Google Scholar 

  26. Mykland, P.A.: Aymptotic expansions and bootstrapping distributions for dependent variables: A martingale approach. Ann. Statist. 20, 623–654 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. McKean, H.P., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom. 1, 43–69 (1967)

    MATH  MathSciNet  Google Scholar 

  28. Protter, P., Talay, D.: The Euler scheme for Levy driven stochastic differential equations. Ann. Probab. 25, 393–323 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. Skorohod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley, Reading [English translation of Skorohod A. V. (1961). Issledovaniya po teorii sluchainykh processov. Kiev University Press] (1965)

  30. Stroock, D.W., Varadhan, S.R.: Multidimensional Diffusion Processes. Springer, Berlin (1979)

    MATH  Google Scholar 

  31. Yoshida, N.: Partial mixing and Edgeworth expansion. Probab. Theory Relat. Fields 129, 559–624 (2004)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Central Economics Mathematical Institute, Academy of Sciences, Nahimovskii av. 47, 117418, Moscow, Russia

    Valentin Konakov

  2. Department of Economics, University of Mannheim, L 7, 3-5, 68131, Mannheim, Germany

    Enno Mammen

Authors
  1. Valentin Konakov
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  2. Enno Mammen
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Corresponding author

Correspondence to Enno Mammen.

Additional information

This research was supported by grant 436RUS113/467/81-2 from the Deutsche Forschungsgemeinschaft and by grants 05-01-04004 and 04-01-00700 from the Russian Foundation of Fundamental Researches. The first author worked on the paper during a visit at the Laboratory of Probability Theory and Random Models of the University Paris VI in 2006. He is grateful for the hospitality during his stay. We would like to thank Stephane Menozzi, two referees and the associate editor for helpful comments

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Konakov, V., Mammen, E. Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains. Probab. Theory Relat. Fields 143, 137–176 (2009). https://doi.org/10.1007/s00440-007-0123-9

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  • Received: 14 December 2006

  • Revised: 29 October 2007

  • Published: 06 December 2007

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0123-9

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Keywords

  • Markov chains
  • Diffusion processes
  • Transition densities
  • Edgeworth expansions

Mathematics Subject Classification (2000)

  • Primary: 62G07
  • Secondary: 60G60
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