• Bernoulli
  • Volume 11, Number 5 (2005), 933-948.

On invariant distribution function estimation for continuous-time stationary processes

Dominique Dehay

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This paper is concerned with the asymptotic behaviour of the empirical distribution function for a large class of continuous-time weakly dependent stationary processes. Under mild mixing conditions the empirical distribution function is an unbiased consistent estimator of the marginal distribution function of the process. For strongly mixing processes this estimator is asymptotically normal. We propose a consistent estimator of the asymptotic variance, and then study the functional central limit theorem for the empirical distribution function.

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Bernoulli, Volume 11, Number 5 (2005), 933-948.

First available in Project Euclid: 23 October 2005

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asymptotic normality central limit theorem consistency continuous time empirical distribution function mixing condition stationary process weak convergence


Dehay, Dominique. On invariant distribution function estimation for continuous-time stationary processes. Bernoulli 11 (2005), no. 5, 933--948. doi:10.3150/bj/1130077600.

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  • [1] Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.
  • [2] Billingsley, P. (1995) Probability and Measure, 3rd edition. New York: Wiley.
  • [3] Castellana, J.V. and Leadbetter, M.R. (1986) On smoothed probability density estimation for stationary process. Stochastic Process Appl., 21, 179-193.
  • [4] Davydov, Yu. (2001) On convergence of empirical measures. Statist. Inference Stochastic Process., 4(1), 1-15.
  • [5] Dehay, D. and Kutoyants, Yu.A. (2004) On confidence intervals for distribution function and density of ergodic diffusion process. J. Statist. Plann. Inference, 124(1), 63-73.
  • [6] Dehling, H., Mikosch, T. and Sørensen, M (2002) Empirical Process Techniques for Dependent Data. Boston: Birkhäuser.
  • [7] Doukhan, P. (1984) Mixing: Properties and Examples, Lecture Notes in Statist. 85. Berlin: Springer- Verlag.
  • [8] Dudley, R.M. (1984) A course on empirical processes. In P.L. Hennequin (ed.), Ecole d´É té de Probabilités de Saint Flour XII-1982, Lecture Notes in Math. 1097, pp. 1-142. Berlin: Springer- Verlag.
  • [9] Ibragimov, I.A. and Linnik, Yu.V. (1971) Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff.
  • [10] Kutoyants, Yu.A. (2003) Statistical Inference for Ergodic Diffusion Processes. London: Springer- Verlag.
  • [11] Meyn, S.P. and Tweedie, R.L. (1993a) Stability of Markovian processes II: Continuous-time processes and sampled chains. Adv. Appl. Probab., 25, 487-517.
  • [12] Meyn, S.P. and Tweedie, R.L. (1993b) Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab., 25, 518-548.
  • [13] Negri, I. (1998) Stationary distribution function estimation for ergodic diffusion process. Statist. Inference Stochastic Process., 1(1), 61-84.
  • [14] Pollard, D. (1990) Empirical Processes: Theory and Applications, NSF-CBMS Reg. Conf. Ser. Probab. Statist. 2. Hayward, CA, and Alexandrias, VA: Institute of Mathematical Statistics and American Statistical Association.
  • [15] Rio, E. (2000) Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants, Math. Appl. 31. Berlin: Springer-Verlag.
  • [16] Shorack, G.R. and Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. New York: Wiley.
  • [17] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. New York: Springer-Verlag.
  • [18] Veretennikov, A.Yu. (1988) Bounds for the mixing rate in the theory of stochastic equations. Theory Probab. Appl., 32(2), 273-281.
  • [19] Veretennikov, A.Yu. (1999) On Castellana-Leadbetter´s condition for diffusion density estimation. Statist. Inference Stochastic Process., 2(1), 1-9.