A NORMAL LIMIT-THEOREM FOR MOMENT SEQUENCES

Let LAMBDA be the set of probability measures lambda on [0, 1]. Let M(n) = {(c1,..., c(n))\lambda is-an-element-of LAMBDA), where c(k) = c(k)(lambda) = integral-1/0x(k) dlambda, k = 1, 2,... are the ordinary moments, and assign to the moment space M(n) the uniform probability measure P(n). We show that, as n --> infinity, the fixed section (c1,..., c(k)), properly normalized, is asymptotically normally distributed. That is, square-root n[(c1,..., c(k)) - (c1(0),..., c(k)0] converges to MVN(0, SIGMA), where c(i)0 correspond to the arc sine law lambda0 on [0, 1]. Properties of the k x k matrix SIGMA are given as well as some further discussion.