Summary
We prove local asymptotic normality (resp. local asymptotic mixed normality) of a statistical experiment, when the observation is a positive-recurrent (resp. null-recurrent, with an additional technical assumption) Markov chain or Markov step process, under rather mild regularity assumptions on the transition kernel for Markov chains, on the infinitesimal generator for Markov processes. The proof makes intensive use of Hellinger processes, thus avoiding almost completely to study the more complicated structure of the likelihoods themselves.
References
Aldous, D.J.: Stopping times and tightness. Ann. Probab.6, 335–340 (1978)
Azéma, J., Duflo, M., Revuz, D.: Mesures invariantes des processus de Markov récurrents. Séminaire de Probabilités III (Lect. Notes Math., vol. 88, pp. 24–33). Berlin Heidelberg New York: Springer 1969
Bingham, N.H.: Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.17, 1–22 (1971)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge: Cambridge University Press 1987
Darling, D.A., Kac, M.: On occupation times for Markoff processes. Trans. Am. Math. Soc.84, 444–458 (1957)
Dzhaparidze, K., Valkeila, E.: On the Hellinger type distances for filtered experiments. Rep. MS-R 8818, CWI, Amsterdam, 1988
Höpfner, R.: Asymptotic inference for continuous-time Markov chains. Probab. Th. Rel. Fields77, 537–550 (1988)
Höpfner, R.: Null recurrent birth-and-death processes, limits of certain martingales, and local asymptotic mixed normality. (preprint 1988, to appear in Scand. J. Stat.)
Höpfner, R.: On limits of some martingales arising in recurrent Markov chains. (preprint 1988)
Ibragimov, I.A., Has'minskii, R.Z.: Statistical estimation, asymptotic theory. Berlin Heidelberg New York: Springer 1981
Jacod, J.: Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 235–253 (1975)
Jacod, J.: Convergence of filtered statistical models and Hellinger processes. Stochastic Processes Appl.32, 47–68 (1989)
Jacod, J.: Une application de la topologie d'Emery: le processus information d'un modèle statistique filtré. Séminaire de Probabilités XXIII (Lect. Notes Math., vol. 1372, pp. 448–474) Berlin Heidelberg New York: Springer 1989
Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin Heidelberg New York: Springer 1987
Kasahara, Y.: A limit theorem for slowly increasing occupation times. Ann. Probab.10, 728–736 (1982)
LeCam, L.: Asymptotic methods in statistical decision theory. Berlin Heidelberg New York: Springer 1986
Maaouia, F.: Thèse de 3ème cycle. Université Paris-7, 1987
Milhaud, X., Oppenheim, G., Viano, M.C.: Sur la convergence du processus de vraisemblance en variables markoviennes. Z. Wahrscheinlichkeitstheor. Verw. Geb.64, 49–65 (1983)
Ogata, Y., Inagaki, N.: The weak convergence of the likelihood ratio random fields for Markov observations. Ann. Inst. Stat. Math.29 A, 165–187 (1977)
Resnick, S., Greenwood, P.: A bivariate stable characterization and domains of attraction. J. Multivariate Anal.9, 206–211 (1979)
Revuz, D.: Markov chains. Amsterdam: North-Holland 1984
Roussas, G.: Contiguity of probability measures. London: Cambridge University Press 1972
Strasser, H.: Mathematical Theory of Statistics. Amsterdam: de Gruyter 1985
Touati, A.: Thèorèmes limites pour des processus de Markov récurrents. (Preprint 1988, to appear in Probab. Th. Rel. Fields). See also C.R. Acad. Sci. Paris Série I305, 841–844 (1987)
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Höpfner, R., Jacod, J. & Ladelli, L. Local asymptotic normality and mixed normality for Markov statistical models. Probab. Th. Rel. Fields 86, 105–129 (1990). https://doi.org/10.1007/BF01207516
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DOI: https://doi.org/10.1007/BF01207516
Keywords
- Statistical Model
- Markov Chain
- Stochastic Process
- Probability Theory
- Statistical Experiment