Limit theorems for a class of identically distributed random variables

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A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (Xₙ)ₙ≥ ₁ is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (Gₙ)ₙ≥ ₀, if it is adapted to (Gₙ)ₙ≥ ₀ and, for each n≥ 0, (Xₖ)ₖ>ₙ is identically distributed given the past Gₙ. In case G₀=,Ω and Gₙ=σ(X₁,...,Xₙ), a result of Kallenberg implies that (Xₙ)ₙ≥ 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 (Xₙ)ₙ≥ ₁ is exchangeable if and only if (Xτ₍ₙ₎)ₙ≥ ₁ is c.i.d. for any finite permutation τof 1,2,..., and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-σ-field. Moreover, (1/n)∑ₖ₌₁ⁿXₖ converges a.s. and in L¹ whenever (Xₙ)ₙ≥ ₁ is (real-valued) c.i.d. and E[| X₁|]<∞. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is E[Xₙ₊₁| Gₙ]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.