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, (Xn) is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (Gn), if it is adapted to (Gn) and, for each n ≥ 0, (Xk:k>n) is identically distributed given the past Gn. In case G0 is trivial and Gn = σ(X1, . . . , Xn), a result of Kallenberg implies that (Xn) 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) is exchangeable if and only if (Xτ(n)) is c.i.d. for any finite permutation τ of {1, 2, . . .}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-σ -field. Moreover, a SLLN and three different CLT's are provided. For each type of CLT, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.