A Normal Limit Theorem for Moment Sequences

Abstract

Let A be the set of probability measures A on [0,1]. Let Mn= {(c1...,cn)tA E A}, where Ck = c_(A) = loXk dA, k = 1, 2, ... are the ordinary moments, and assign to the moment space Mn the uniform probability measure Pn. We show that, as n -oo, the fixed section (cl,...Ck), properly normalized, is asymptotically normally distributed. That is, Vn [(c1,. ., Cd) (c?.. Ckr)] converges to MVN(O, 1), where c? correspond to the arc sine law AO on [0, 1]. Properties of the k x k matrix X are given as well as some further discussion.