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)n greater than or equal to 1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (g(n))(n greater than or equal to 0), if it is adapted to (g(n))(ngreater than or equal to 0) and, for each n greater than or equal to 0, (X-k)(k>n) is identically distributed given the past g(n). In case g(0) is trivial and g(n) = sigma(X-1,...,X-n), a result of Kallenberg implies that (X-n)(n greater than or equal to 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 greater than or equal to 1) is exchangeable if and only if (X-tau(n))(n greater than or equal to 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) Sigma(k=1)(n) X-k converges a. s. and in L-1 whenever (X-n)(n greater than or equal to 1) is (real-valued) c.i.d. and E[|X-1|]