On the central limit theorem for dynamical systems

Authors:
Robert Burton and Manfred Denker

Journal:
Trans. Amer. Math. Soc. **302** (1987), 715-726

MSC:
Primary 60F05; Secondary 28D05

DOI:
https://doi.org/10.1090/S0002-9947-1987-0891642-6

MathSciNet review:
891642

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Abstract: Given an aperiodic dynamical system $(X,T,\mu )$ then there is an $f \in {L^2}(\mu )$ with $\smallint fd\mu = 0$ satisfying the Central Limit Theorem, i.e. if ${S_m}f = f + f \circ T + \cdots + f \circ {T^{m - 1}}$ and ${\sigma _m} = {\left \| {{S_m}f} \right \|_2}$ then \[ \mu \left \{ {x|\frac {{{S_m}f(x)}}{{{\sigma _m}}} < u} \right \} \to {(2\pi )^{ - 1/2}}\int _{ - \infty }^u {{\text {exp}}} \left [ {\frac {{ - {\upsilon ^2}}}{2}} \right ]d\upsilon .\] The analogous result also holds for flows.

- R. N. Bhattacharya,
*On the functional central limit theorem and the law of the iterated logarithm for Markov processes*, Z. Wahrsch. Verw. Gebiete**60**(1982), no. 2, 185–201. MR**663900**, DOI https://doi.org/10.1007/BF00531822 - Patrick Billingsley,
*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396** - Manfred Denker and Michael Keane,
*Almost topological dynamical systems*, Israel J. Math.**34**(1979), no. 1-2, 139–160 (1980). MR**571401**, DOI https://doi.org/10.1007/BF02761830 - Manfred Denker and Walter Philipp,
*Approximation by Brownian motion for Gibbs measures and flows under a function*, Ergodic Theory Dynam. Systems**4**(1984), no. 4, 541–552. MR**779712**, DOI https://doi.org/10.1017/S0143385700002637
D. Dürr and S. Goldstein, - M. I. Gordin and B. A. Lifšic,
*Central limit theorem for stationary Markov processes*, Dokl. Akad. Nauk SSSR**239**(1978), no. 4, 766–767 (Russian). MR**0501277** - Franz Hofbauer and Gerhard Keller,
*Ergodic properties of invariant measures for piecewise monotonic transformations*, Math. Z.**180**(1982), no. 1, 119–140. MR**656227**, DOI https://doi.org/10.1007/BF01215004 - I. A. Ibragimov,
*Some limit theorems for stationary processes*, Teor. Verojatnost. i Primenen.**7**(1962), 361–392 (Russian, with English summary). MR**0148125** - Gisiro Maruyama,
*Nonlinear functionals of Gaussian stationary processes and their applications*, Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975) Springer, Berlin, 1976, pp. 375–378. Lecture Notes in Math., Vol. 550. MR**0433575** - C. M. Newman and A. L. Wright,
*An invariance principle for certain dependent sequences*, Ann. Probab.**9**(1981), no. 4, 671–675. MR**624694** - Walter Philipp and William Stout,
*Almost sure invariance principles for partial sums of weakly dependent random variables*, Mem. Amer. Math. Soc.**2**(1975), no. 161,, 161, iv+140. MR**433597**, DOI https://doi.org/10.1090/memo/0161 - M. Ratner,
*The central limit theorem for geodesic flows on $n$-dimensional manifolds of negative curvature*, Israel J. Math.**16**(1973), 181–197. MR**333121**, DOI https://doi.org/10.1007/BF02757869
A. Rényi, - M. Rosenblatt,
*A central limit theorem and a strong mixing condition*, Proc. Nat. Acad. Sci. U.S.A.**42**(1956), 43–47. MR**74711**, DOI https://doi.org/10.1073/pnas.42.1.43
A. Rothstein, Personal communication, 1984.
- R. Salem and A. Zygmund,
*On lacunary trigonometric series*, Proc. Nat. Acad. Sci. U.S.A.**33**(1947), 333–338. MR**22263**, DOI https://doi.org/10.1073/pnas.33.11.333

*Remarks and the central limit theorem for weakly dependent sequences*, Preprint, 1985.

*Wahrscheinlichkeitrechnung*, VEB Deutscher Verlag Wiss., Berlin, 1962.

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Keywords:
Central limit theorem,
dynamical systems

Article copyright:
© Copyright 1987
American Mathematical Society