Abstract
This paper considers the problem of estimation of drift parameter for linear homogeneous stochastic difference equations. The Local Asymptotic Normality (LAN) for the problem is proved. LAN implies the Hajek–Le Cam minimax lower bound. In particular, it is shown that the Fisher's information matrix for the problem can be expressed in terms of the stationary distribution of an auxiliary Markov chain on the projective space P(ℝd).
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REFERENCES
Apostol, T. M. (1957). Mathematical Analysis, Addison-Wesley, Reading, London.
Bougerol, P., and Lacroix, J. (1985). Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, Boston/Basel/Stuttgart.
Crisanti, A., Paladin, G., and Vulpiani, A. (1993). Products of Random Matrices in Statistical Physics, Springer, Berlin/Heidelberg/New York.
Doob, J. L. (1953). Stochastic Processes, John Wiley, New York, Chapman & Hall, London.
Furstenberg, H. (1963). Noncommuting random products. Trans. Amer. Math. Soc. 108, 377–428.
Furstenberg, H., and Kesten, H. (1960). Products of random matrices. Ann. Math. 31, 457–469.
Goldsheid, I. Ya., and Margulis, G. A. (1989). Lyapunov indices of a product of random matrices. Russian Math. Surveys 44(5), 11–71.
Ibragimov, I. A., and Khasminskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory, Springer-Verlag, Berlin.
Ibragimov, I. A., and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, the Netherlands.
Jankunas, A., and Khasminskii, R. Z. (1997). Estimation of parameters of linear homogeneous stochastic differential equations. Stochastic Processes and Their Applications 72(2).
Jankunas, A., and Khasminskii, R. Z. (1997). Estimation of parameters of linear stochastic difference equations. Math. Methods Stat. (submitted).
Khasminskii, R. Z. (1980). Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands.
Khasminskii, R. Z., Krylov, N., and Moshchuk, N. (1997). Estimation of parameters of linear stochastic differential equations with singular diffusion. Prob. Th. Rel. Fields (submitted).
Roerdnik, J. B. T. M. (1988). The biennial life strategy in a random environment. J. Math. Biol. 26, 199–215.
Tuljapurkar, S. D., and Orzack, S. T. (1980). Population dynamics in variable environments. 1. Long-run growth rates and extinction. Theor. Popul. Biol. 18, 314–342.
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Jankunas, A. Local Asymptotic Normality for Linear Homogeneous Difference Equations with Non-Gaussian Noise. Journal of Theoretical Probability 12, 675–697 (1999). https://doi.org/10.1023/A:1021623714825
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DOI: https://doi.org/10.1023/A:1021623714825