A central limit theorem of mixed type for a class of 1-dimensional transformations
Let T be a nonsingular transformation on the interval [0,1]. Many authors have investigated the following problem: under what conditions on T and f does the sequence of random variables\(\{f(T^ kx):\quad k=0,1,...\}\) satisfy a central limit theorem (c.l.t.)? Recently, J. Rousseau-Egele obtained a c.l.t. for a class of transformations T and its rate of convergence, by estimating the asymptotic behavior of the characteristic function with the help of the Perron-Frobenius operator corresponding to T. Generalizing his method, we can obtain central limit theorems of mixed type for a certain class of transformations. That is, under suitable assumptions on T, f and \(\nu\), the distribution function \(\nu \{\sum^{n-1}_{k=0}f(T^ kx)/n^{1/2}<z\}\) is asymptotically a mixed normal distribution function. Central limit theorems for \(\beta\)- transformations, \(\alpha\)-continued fraction transformations, Wilkinson's piecewise linear transformations and unimodal linear transformations are given as corollaries to our theorems.
scientific article; zbMATH DE number 4076184