CENTRAL LIMIT THEOREM ON CHEBYSHEV POLYNOMIALS

Abstract

Let <TEX>$T_l$</TEX> be a transformation on the interval [-1, 1] defined by Chebyshev polynomial of degree <TEX>$l(l{\geq}2)$</TEX>, i.e., <TEX>$T_l(cos{\theta})=cos(l{\theta})$</TEX>. In this paper, we consider <TEX>$T_l$</TEX> as a measure preserving transformation on [-1, 1] with an invariant measure <TEX>$\frac{1}{\sqrt[\pi]{1-x^2}}dx$</TEX>. We show that If f(x) is a nonconstant step function with finite k-discontinuity points with k < l-1, then it satisfies the Central Limit Theorem. We also give an explicit method how to check whether it satisfies the Central Limit Theorem or not in the cases of general step functions with finite discontinuity points.