A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_{n\geq 1} is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\geq 0}, if it is adapted to (G_n)_{n\geq 0} and, for each n\geq 0, (X_k)_{k>n} is identically distributed given the past G_n. In case G_0={\varnothing,\Omega} and G_n=\sigma(X_1,...,X_n), a result of Kallenberg implies that (X_n)_{n\geq 1} is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X_n)_{n\geq 1} is exchangeable if and only if (X_{\tau(n)})_{n\geq 1} is c.i.d. for any finite permutation \tau of {1,2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-\sigma-field. Moreover, (1/n)\sum_{k=1}^nX_k converges a.s. and in L^1 whenever (X_n)_{n\geq 1} is (real-valued) c.i.d. and E[| X_1| ]<\infty. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is E[X_{n+1}| G_n]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000067

Let (X n ) be a sequence of integrable real random variables, adapted to a filtration (G n ).stably are given, where U and V are certain random variables. In particular, under such conditions, we obtainstably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

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In a Bayesian framework, to make predictions on a sequence X 1 , X 2 , . . . of random observations, the inferrer needs to assign the predictive distributions σ n (•) = P (X n+1 ∈ • | X 1 , . . . , X n ). In this paper, we propose to assign σ n directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be assessed. The data sequence (X n ) is assumed to be conditionally identically distributed (c.i.d.) in the sense of (Ann. Probab. 32 (2004) 2029-2052). To realize this programme, a class of predictive distributions is introduced and investigated. Such a is rich enough to model various real situations and (X n ) is actually c.i.d. if σ n belongs to . Furthermore, when a new observation X n+1 becomes available, σ n+1 can be obtained by a simple recursive update of σ n . If μ is the a.s. weak limit of σ n , conditions for μ to be a.s. discrete are provided as well.

Let (Ω, B, P ) be a probability space, A ⊂ B a sub-σ-field, and µ a regular conditional distribution for P given A. Necessary and sufficient conditions for µ(ω)(A) to be 0-1, for all A ∈ A and ω ∈ A0, where A0 ∈ A and P (A0) = 1, are given. Such conditions apply, in particular, when A is a tail sub-σ-field. Let H(ω) denote the Aatom including the point ω ∈ Ω. Necessary and sufficient conditions for µ(ω)(H(ω)) to be 0-1, for all ω ∈ A0, are also given. If (Ω, B) is a standard space, the latter 0-1 law is true for various classically interesting sub-σ-fields A, including tail, symmetric, invariant, as well as some sub-σ-fields connected with continuous time processes.