@article{86af2d5c7150488b9e6976933886a949,

title = "Asymptotically Gaussian distribution for random perturbations of rotations of the circle",

abstract = "Let Tε,ω be a transformation of the two-dimensional torus T2 given by the formula Tε,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 ∑k=0∞ f o Tε,ω k, where ε is an irrational number and f belongs to a class of real-valued functions on T2 described in terms of ε. The proof will be published elsewhere.",

author = "M. Denker and M. Gordin",

note = "Funding Information: It is obvious that (3) follows from (5), and (4) follows from (5) and (6). We note that (5), in turn, is satisfied for any irrational algebraic ~ (a theorem by Thue--Siegel-Roth, see \[4\])a, nd also for almost all ~ G IR (as a consequence of a stronger measure-theoretical result in Diophantine approximation theory \[4\]). Under the assumptions of the theorem stated above, a more detailed conclusion can also be made which shows that the separate harmonics in the expansion f(z, y) = ~kEZ fk(x) exp(2~riky) give independent contributions to the limiting Gaussian distribution. The precise statement of this assertion and the proof will be published elsewhere. 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 l~ussian Foundation for Basic t~esearch, grant 96-01-00672.",

year = "1999",

doi = "10.1007/BF02175827",

language = "English (US)",

volume = "96",

pages = "3493--3495",

journal = "Journal of Mathematical Sciences",

issn = "1072-3374",

publisher = "Springer Science and Business Media Deutschland GmbH",

number = "5",

}