n - a

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 µn be a probability measure on the Borel sigma-field on D[0, 1] with respect to Skorohod distance, n = 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables Xn such that Xn tilde µn for all n =...

Empirical processes for non ergodic data are investigated under uniform distance. Some CLTs, both uniform and non uniform, are proved. In particular, conditions for Bn = n^(1/2) (µn - bn) and Cn = n^(1/2) (µn - an) to converge in distribution are given, where µn is the empirical measure, an...

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

Let L be a linear space of real bounded random variables on the probability space (omega,A, P0). There is a finitely additive probability P on A, such that P tilde P0 and EP (X) = 0 for all X in L, if and only if cEQ(X) = ess sup(-X), X in L, for some constant c 0 and (countably additive)...

Let (µn : n = 0) be Borel probabilities on a metric space S such that µn - µ0 weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables Xn satisfying Xn - µn for all n and Xn - X0 in probability. By Skorohod's theorem, Skorohod...

Let (omega,F,P) be a probability space. For each G in F, define G as the s-field generated by G and those sets f in F satisfying P(f) in {0, 1}. Conditions for P to be atomic on the intersection of the complements of Ai for i=1,..,k, with A1, . . . ,Ak in F sub-s-fields, are given. Conditions...

This paper deals with empirical processes of the type Cn(B) = n^(1/2) {µn(B) - P(Xn+1 in B