Asymptotic predictive inference with exchangeable data

Let (X-n) be a sequence of random variables, adapted to a filtration (G(n)), and let mu(n) = (1/n) Sigma(n)(i=1) X-delta(i) and a(n)(.) = P(Xn+ 1 is an element of.vertical bar G(n)) be the empirical and the predictive measures. We focus on parallel to mu(n)-a(n)parallel to = sup(B is an element of D)vertical bar mu(n)(B)-a(n)(B)vertical bar, where D is a class of measurable sets. Conditions for parallel to mu(n) - a(n) parallel to -> 0, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of r(n) parallel to mu(n) -a(n)parallel to is investigated for suitable constants r(n). Special attention is paid to r(n) = root n and r(n) = root n/log log n. The sequence (X-n) is exchangeable or, more generally, conditionally identically distributed.