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)].
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) |
References:
| [1] | Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms (1975), Berlin: Springer-Verlag, Berlin · Zbl 0308.28010 |
| [2] | Denker, M.; Philipp, W., Approximation by Brownian motion for Gibbs measures and flows under a function, Ergodic Theory & Dynamical Systems, 4, 541-552 (1984) · Zbl 0554.60077 |
| [3] | Feller, W., An Introduction to Probability Theory and its Applications (1971), New York: John Wiley & Sons, New York · Zbl 0219.60003 |
| [4] | Kato, T., Perturbation Theory for Linear Operators (1980), New York: Springer-Verlag, New York · Zbl 0435.47001 |
| [5] | Keller, G., Generalised bounded variation and applications to piecewise monotonic transformations, Z. Wahrscheinlichkeitstheor. Verw. Geb., 69, 461-478 (1985) · 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] | Lalley, S., Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math., 8, 154-193 (1987) · Zbl 0637.58013 · doi:10.1016/0196-8858(87)90012-1 |
| [8] | Pollicott, M., A complex Ruelle-Perron-Frobenius theorem and two counterexamples, Ergodic Theory & Dynamical Systems, 4, 135-146 (1984) · Zbl 0575.47009 |
| [9] | [Po2] M. Pollicott,Prime orbit theorem error terms for locally constant suspensions, preprint (1985). |
| [10] | Ratner, M., The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Isr. J. Math., 16, 181-197 (1973) · Zbl 0283.58010 · doi:10.1007/BF02757869 |
| [11] | Rousseau-Egèle, J., Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. of Prob., 11, 772-788 (1983) · Zbl 0518.60033 · doi:10.1214/aop/1176993522 |
| [12] | Ruelle, D., Thermodynamic Formalism (1978), Reading, Mass.: Addison-Wesley, Reading, Mass. · Zbl 0401.28016 |
| [13] | Schmidt, W., Diophantine Approximation (1980), Berlin: Springer-Verlag, Berlin · Zbl 0421.10019 |
| [14] | Sinai, Y. G., The central limit theorem for geodesic flows on manifolds of constant negative curvature, Soviet Math. Dokl., 1, 983-987 (1960) · Zbl 0129.31103 |
| [15] | Wong, S., A central limit theorem for piecewise monotonic mappings of the interval, Ann. of Prob., 7, 500-514 (1979) · Zbl 0413.60014 · doi:10.1214/aop/1176995050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
