Let (Xn) be a sequence of integrable real random variables, adapted to a filtration (Gn). Define: Cn = n^(1/2) {1/n SUM(k=1:n) Xk - E(Xn+1 | Gn) } and Dn = n^(1/2){ E(Xn+1 | Gn)-Z } where Z is the a.s. limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn,Dn) -- N(0,U) × N(0,V) stably...

An urn contains balls of d = 2 colors. At each time n = 1, a ball is drawn and then replaced together with a random number of balls of the same color. Let An =diag (An,1, . . . ,An,d) be the n-th reinforce matrix. Assuming EAn,j = EAn,1 for all n and j, a few CLT’s are available for such urns....

Let (omega, beta) be a measurable space, An in B a sub-sigma-field and µn a random probability measure, n = 1. In various frameworks, one looks for a probability P on B such that µn is a regular conditional distribution for P given An for all n. Conditions for such a P to exist are given. The...

Let S be a Polish space and (Xn : n = 1) an exchangeable sequence of S-valued random variables. Let an(·) = P( Xn+1 in · | X1, . . . ,Xn) be the predictive measure and a a random probability measure on S such that an (weak) -- a a.s.. Two (related) problems are addressed. One is to give...