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Central limit asymptotics for shifts of finite type. (English) Zbl 0701.60026
The authors prove the Berry-Esséen bound in the central limit theorem for Hölder-continuous functions on a shift of finite type endowed with a Gibbs measure [cf. Y. Guivarc’h and J. Hardy, Ann. Inst. Henri Poincaré, Probab. Stat. 24, 73-98 (1988; Zbl 0649.60041)]. Moreover, if f has a non lattice distribution the asymptotic expansion up to order o(1/$$\sqrt{n})$$ is determined, and under certain moment conditions higher order approximations are derived.
The method of proof is based on the theory of Perron-Frobenius operators [cf. J. Rousseau-Egele, Ann. Probab. 11, 772-788 (1983; Zbl 0518.60033)].
Reviewer: M.Denker

##### MSC:
 60F99 Limit theorems in probability theory 28D20 Entropy and other invariants 60K35 Interacting random processes; statistical mechanics type models; percolation theory 54H20 Topological dynamics (MSC2010)
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##### References:
 [1] [Bo] R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math.470, Springer-Verlag, Berlin, 1975. · Zbl 0308.28010 [2] [DP] M. Denker and W. Philipp,Approximation by Brownian motion for Gibbs measures and flows under a function, Ergodic Theory & Dynamical Systems4 (1984), 541–552. · Zbl 0554.60077 [3] [Fe] W. Feller,An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., John Wiley & Sons, New York, 1971. · Zbl 0219.60003 [4] [Ka] T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980. [5] [Ke] G. Keller,Generalised bounded variation and applications to piecewise monotonic transformations. Z. Wahrscheinlichkeitstheor. Verw. Geb.69 (1985), 461–478. · Zbl 0574.28014 · doi:10.1007/BF00532744 [6] [La1] S. Lalley,Ruelle’s Perron-Frobenius theorem and central limit theorem for additive functionals of one-dimensional Gibbs states, Proc. Conf. in honour of H. Robbins, 1985. [7] [La2] S. Lalley,Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math.8 (1987), 154–193. · Zbl 0637.58013 · doi:10.1016/0196-8858(87)90012-1 [8] [Po1] M. Pollicott,A complex Ruelle-Perron-Frobenius theorem and two counterexamples, Ergodic Theory & Dynamical Systems4 (1984), 135–146. · Zbl 0575.47009 [9] [Po2] M. Pollicott,Prime orbit theorem error terms for locally constant suspensions, preprint (1985). [10] [Ra] M. Ratner,The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Isr. J. Math.16 (1973), 181–197. · Zbl 0283.58010 · doi:10.1007/BF02757869 [11] [R-E] J. Rousseau-Egèle,Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. Ann. of Prob.11 (1983), 772–788. · Zbl 0518.60033 · doi:10.1214/aop/1176993522 [12] [Ru] D. Ruelle,Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978. [13] [Sc] W. Schmidt,Diophantine Approximation, Springer Lecture Notes in Math.785, Springer-Verlag, Berlin, 1980. [14] [Si] Y. G. Sinai,The central limit theorem for geodesic flows on manifolds of constant negative curvature, Soviet Math. Dokl.1 (1960), 983–987. · Zbl 0129.31103 [15] [Wo] S. Wong,A central limit theorem for piecewise monotonic mappings of the interval, Ann. of Prob.7 (1979), 500–514. · Zbl 0413.60014 · doi:10.1214/aop/1176995050
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