TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 302, Number 2, August 1987 ON THE CENTRAL LIMIT THEOREM FOR DYNAMICAL SYSTEMS ROBERT BURTON AND MANFRED DENKER ABSTRACT. Given an aperiodic dynanlical system (X, T, p,) then there is an ! E L2{p,) with J ! dp, = 0 satisfying the Central Limit Theorem, i.e. if Sm! = ! +! 0 T + ... +! 0 Tm-1 and Urn = IISm!112 then p, {xI < u} {21T)-1/2 exp [ _;2] dv. The analogous result also holds for flows. 1. Introduction. The most important and most studied theorem in probability theory is the Central Limit Theorem (CLT) which may be stated in the context of dynamical systems. If (X, T, J.L) is a dynamical system (i.e. T is a measurable, measure-preserving transformation of the Lebesgue probability space (X,J.L)) and if I E L2 (J.L) is centered, i.e. J x I dJ.L = 0, with I, loT, I 0 T2, . .. forming an independent sequence then (1) where 8m l = I + loT + ... + f 0 rm- 1 , II 8 m III = L2-norm of 8m l = standard deviation of 8 m /, and = 2(11')-1/2 exp[-v 2 /2] dv. In this case 118 m /ll = Vmii7if but we shall drop the independent assumption above, and we shall then say that IE L 2 (J.L) satisfies the CLT if (1) holds. Many generalizations of the CLT are in the literature where (1) is shown to hold for a wider class of functions. To the authors' knowledge, all of these work by weakening the independence assumptions. As examples, (1) still holds for certain functions I where I, loT, ... satisfy mixing conditions of Rosenblatt or Ibragimov [8, 14] or form a martingale [2] or satisfy positive dependence conditions with summable convariance functions as in Newman and Wright [10]. Some attention has been paid to CLT for special dynamical systems, such as Ratner [12], Denker and Philipp [4], Hofbauer and Keller [7], and others cited there. All these results are deduced from kinds of mixing as described above. Similar things can be said about flows, built under a function, but the situation is somewhat more complicated. All these theorems and examples imply at least the K-property, so far no CLT for a dynamical system seems to be published where Received by the editors September 5, 1985 and, in revised form, July 28, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 60F05j Secondary 28D05. Key words and phrases. Central limit theorem, dynanlical systems. The first author was supported in part by the Alexander von Humboldt Foundation and NSF grant #MCS-800 5172. 715 ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use