Invariance principles and Gaussian approximation for strictly stationary processes

Volný, Dalibor

doi:10.1090/S0002-9947-99-02401-0

Invariance principles and Gaussian approximation for strictly stationary processes
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by Dalibor Volný

Trans. Amer. Math. Soc. 351 (1999), 3351-3371

DOI: https://doi.org/10.1090/S0002-9947-99-02401-0

Published electronically: April 8, 1999

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Abstract:

We show that in any aperiodic and ergodic dynamical system there exists a square integrable process $(f\circ T^{i})$ the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For $(f\circ T^{i})$ both weak and strong invariance principles hold.

References

Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
Robert Burton and Manfred Denker, On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc. 302 (1987), no. 2, 715–726. MR 891642, DOI 10.1090/S0002-9947-1987-0891642-6
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
N. I. Chernov, Limit theorems and Markov approximations for chaotic dynamical systems, Probab. Theory Related Fields 101 (1995), no. 3, 321–362. MR 1324089, DOI 10.1007/BF01200500
M. Csörgő and P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 666546
De La Rue, T., Ladouceur, S., Peškir, G., and Weber, M., On the central limit theorem for aperiodic dynamical systems and applications, preprint 1993.
Manfred Denker and Michael Keane, Almost topological dynamical systems, Israel J. Math. 34 (1979), no. 1-2, 139–160 (1980). MR 571401, DOI 10.1007/BF02761830
M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741 (Russian). MR 0251785
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Keane, M. and Volný, D., an unpublished result.
Michael T. Lacey, On weak convergence in dynamical systems to self-similar processes with spectral representation, Trans. Amer. Math. Soc. 328 (1991), no. 2, 767–778. MR 1066446, DOI 10.1090/S0002-9947-1991-1066446-5
Michael T. Lacey, On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, J. Anal. Math. 61 (1993), 47–59. MR 1253438, DOI 10.1007/BF02788838
Lesigne, E., Almost sure central limit theorem for strictly stationary processes, preprint 1996.
Lesigne, E., personal communication, 1996.
Liardet, P. and Volný, D., Sums of continuous and differentiable functions in dynamical systems, Israel J. of Mathematics 98 (1997), 29-60.
Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631
Thouvenot, J.-P. and Weiss, B., personal communication.
Dalibor Volný, On limit theorems and category for dynamical systems, Yokohama Math. J. 38 (1990), no. 1, 29–35. MR 1093661
Dalibor Volný, Approximating martingales and the central limit theorem for strictly stationary processes, Stochastic Process. Appl. 44 (1993), no. 1, 41–74. MR 1198662, DOI 10.1016/0304-4149(93)90037-5