Abstract
Let T ε,ω be a transformation of the two-dimensional torus T 2 given by the formula T ε,ω ∶ (x, y)→(2x, y+ Ω+εx) mod 1.A version of the functional central limit theorem is formulated for variables of the form \(n^{ - 1/2} \sum\nolimits_{k = 0}^\infty {f o T_{\varepsilon ,\omega }^k } \),where ε is an irrational number and f belongs to a class of real- valued functions on T 2 described in terms of ε. The proof will be published elsewhere.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Borodin and I. A. Ibragimov,Limit Theorems for Functionals of Random Walks, AMS (1995).
M. I. Gordin, “On the central limit theorem for stationary processes,”Sov. Math. Dokl.,10, No. 4, 1174–1176 (1969).
M. I. Gordin and B. A. Lifshits, “The central limit theorem for stationary Markov processes,”Sov. Math. Dokl.,19, No. 2, 392–394 (1978).
J. W. C. Cassels,An Introduction to Diophantine Approximation, Cambridge University Press (1957).
W. Parry, “Ergodic properties of a one-parameter family of skew-products,”Nonlinearity,8, 821–825 (1995).
W. Parry, “Automorphisms of the Bernoulli endomorphisms and a class of skew-products,”Ergodic Theory and Dynamical Systems,16, 519–529 (1996).
S. Siboni, “Ergodic properties of a class of skew-systems obtained by coupling the translation of the 1-torus with the endomorphism 2x mod 1,”Nonlinearity,7, 1133–1141 (1994).
Additional information
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 78–81.
This work was performed as part of the Russian-German project DFG-RFBR, grant 96-01-00096. The second author was also supported by the Russian Foundation for Basic Research, grant 96-01-00672.
Rights and permissions
About this article
Cite this article
Denker, M., Gordin, M. Asymptotically gaussian distribution for random perturbations of rotations of the circle. J Math Sci 96, 3493–3495 (1999). https://doi.org/10.1007/BF02175827
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02175827