Limit theorems for predictive sequences of random variables

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (Xn)n≥0 is said to be predictive, with respect to a filtration (Gn)n≥0 such that Gn ⊃ σ(X0, . . . , Xn), if X0 is distributed as X1 and, for each n ≥ 0, (Xk)k>n is identically distributed given the past Gn. In case Gn = σ(X0, . . . , Xn), a result of Kallenberg implies that (Xn)n≥0 is exchangeable if and only if is stationary and predictive. After giving some natural examples of non exchangeable predictive sequences, it is shown that (Xn)n≥0 is exchangeable if and only if (Xτ(n))n≥0 is predictive for any finite permutation τ of N, and that the distribution of a predictive sequence agrees with an exchangeable law on a certain sub-σ-field. Moreover, (1/n) ∑n−1 k=0 f(Xk) converges a.s. and in L 1 whenever (Xn)n≥0 is predictive and f is a real measurable function such that E[|f(X0)|] < ∞. As to the CLT , three types of random centering are considered. One of such centerings, significant in Bayesian prediction and discrete time filtering, is E[f(Xn+1)|Gn]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.