The Central Limit Theorem for Piecewise Linear Transformations

The purpose of the present paper is to give central limit theorems for piecewise linear transformations ([6]), which are generalizations of /^-transformations and which belong to a class of number-theoretical transformations with "dependent digits" (cf. [4]). The central limit theorems for ones with "independent digits" are studied by many authors ([!]> [7] etc.). However, the cases of "dependent digits" seem to be not studied. These cases are more complicated than the cases of "independent digits". In [2] it is shown that the ^-transformations have Ornstein's weak Bernoulli property. Then it is easy to see by an analogous way to [2] that our transformations also satisfy the weak Bernoulli condition. Therefore Ornstein and Friedman's theorem implies that the natural extensions of our transformations are isomorphic to the Bernoulli shifts. But we never know how to construct their Bernoulli generators. Hence the classical central limit theorems for the Bernoulli shifts imply no concrete result for our transformations. We modify the method, which is used in [2] to prove the weak Bernoulli property of jS-transforma tions, to show that the natural generators of piecewise linear transformations satisfy Rosenblatt's strong mixing condition. Thus we obtain central limit theorems. By virtue of the good properties of our generators, we obtain concrete results, namely if / is of bounded variation or Holder continuous, we get the central limit