  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.
