Abstract. Let (Xn) be any sequence of random variables adapted to a filtration (Gn). Define an(·) = P ( ) 1 ∑n−1 Xn+1 ∈ · | Gn, bn = n i=0 ai, and µn = 1 ∑n n i=1 δX. Convergence in distribution of the empirical processes i Bn = √ n (µn − bn) and Cn = √ n (µn − an) is investigated under uniform distance. If (Xn) is conditionally identically distributed (in the sense of ) convergence of Bn and Cn is studied according to Meyer-Zheng as well. Some CLTs, both uniform and non uniform, are proved. In addition, various examples and a characterization of conditionally identically distributed sequences are given. 1
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