Limit theorems for a class of identically distributed random variables

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X-n)ngreater than or equal to1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (g(n))(ngreater than or equal to0), if it is adapted to (g(n))(ngreater than or equal to0) and, for each ngreater than or equal to0, (X-k)(k>n) is identically distributed given the past g(n). In case g(0) = {theta, Omega} and g(n) = sigma(X-1,...,X-n), a result of Kallenberg implies that (X-n)(ngreater than or equal to1) 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)(ngreater than or equal to1) is exchangeable if and only if (X-tau(n))(ngreater than or equal to1) 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) Sigma(k=1)(n) X-k converges a. s. and in L-1 whenever (X-n)(ngreater than or equal to1) is (real-valued) c.i.d. and E[\X-1\]<infinity. 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\g(n)]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.