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Edgeworth expansion for ergodic diffusions
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  • Published: 03 August 2007

Edgeworth expansion for ergodic diffusions

  • Masaaki Fukasawa1 

Probability Theory and Related Fields volume 142, pages 1–20 (2008)Cite this article

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Abstract

The Edgeworth expansion for an additive functional of an ergodic diffusion is validated under fairly weak conditions. The validation procedure does not depend on the stationarity or the geometric mixing property, but exploits the strong Markov property of the process. In particular for an Itô-diffusion of dimension one, verifiable conditions for the validity of the expansion are given in terms of the coefficients of the corresponding stochastic differential equation. The maximum likelihood estimator for the CIR process is treated as example.

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References

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertail P. and Clémençon S. (2007). Second-order properties of regeneration-based bootstrap for atomic Markov chains. Test 16: 109–122

    Article  MathSciNet  MATH  Google Scholar 

  4. Bhattacharya R.N. and Ghosh J.K. (1976). On the validity of the formal Edgeworth expansion. Ann. Stat. 6: 434–451

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bolthausen E. (1982). The Berry–Esseén theorem for strongly mixing Harris recurrent Markov chains. Z. Wahr. 60: 283–289

    Article  MathSciNet  MATH  Google Scholar 

  8. Borisov I.S. (1978). Estimate of the rate of convergence of distributions of additive functionals of a sequence of sums of independent random variables. Siberian Math. J. 19(3): 371–383

    Article  Google Scholar 

  9. Fitzsimmons P.J. and Pitman J. (1999). Kac’s moment formula and Feynman–Kac formula for additive functionals of a Markov process. Stoc. Proc. Appl. 79: 117–134

    Article  MathSciNet  MATH  Google Scholar 

  10. Fukasawa, M.: Regenerative block bootstrap for ergodic diffusions (in press)

  11. Gikhman I.I. and Skorokhod A.V. (1972). Stochastic Differential Equations. Springer, Berlin

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Hall P. (1992). The bootstrap and Edgeworth expansion. Springer, New York

    Google Scholar 

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

    MATH  Google Scholar 

  15. Karatzas I. and Shreve S.E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Malinovskii V.K. (1987). Limit theorems for Harris Markov chains, 1. Theory Probab. Appl. 31(2): 269–285

    Article  Google Scholar 

  18. Mykland P.A. (1992). Asymptotic expansions and bootstrapping distributions for dependent variables: a martingale approach. Ann. Stat. 20: 623–654

    Article  MathSciNet  MATH  Google Scholar 

  19. Nummelin E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press, London

    MATH  Google Scholar 

  20. Revuz D. and Yor M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin

    MATH  Google Scholar 

  21. Sakamoto Y. and Yoshida N. (1998). Asymptotic expansion of M-estimator over Wiener space. Stat. Inference Stoch. Process. 1: 85–103

    Article  MathSciNet  MATH  Google Scholar 

  22. Sakamoto Y. and Yoshida N. (2004). Asymtotic expansion formulas for functionals of ε-Markov processes with a mixing property. Ann. Inst. Stat. Math. 56(3): 545–597

    Article  MathSciNet  MATH  Google Scholar 

  23. Skorokhod A.V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations. American Mathematical Society, USA

    MATH  Google Scholar 

  24. Veretennikov A.Yu. (2006). On lower bounds for mixing coefficients of Markov diffusions. In: Stoyanov, Y.M., Kabanov, J.M. and Lipster, R. (eds) From Stochastic Calculus to Mathematical Finance., pp 623–633. Springer, Berlin

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

    Masaaki Fukasawa

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  1. Masaaki Fukasawa
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Correspondence to Masaaki Fukasawa.

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Fukasawa, M. Edgeworth expansion for ergodic diffusions. Probab. Theory Relat. Fields 142, 1–20 (2008). https://doi.org/10.1007/s00440-007-0097-7

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  • Received: 05 March 2006

  • Revised: 20 June 2007

  • Published: 03 August 2007

  • Issue Date: September 2008

  • DOI: https://doi.org/10.1007/s00440-007-0097-7

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Mathematics Subject Classification (2000)

  • 60F05
  • 60J25
  • 60J55
  • 62E20
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