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 mo
ments, and assign to the moment space M(n) the uniform probability mea
sure P(n). We show that, as n --> infinity, the fixed section (c1,...,
c(k)), properly normalized, is asymptotically normally distributed. T
hat 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.