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, $(X_n)_{n\geq 1}$
is said to be conditionally identically distributed (c.i.d.), with
respect to a filtration $(\mathcal{G}_n)_{n\geq 0}$, if it is
adapted to $(\mathcal{G}_n)_{n\geq 0}$ and, for each $n\geq 0$, $(X_k)_{k>n}$ is identically distributed given the past
$\mathcal{G}_n$. In case $\mathcal{G}_0=\{\emptyset,\Omega\}$ and $\mathcal{G}_n=\sigma(X_1,\ldots,X_n)$, 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 non-exchangeable 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 $\{1,2,\ldots\}$, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-$\sigma$-field. Moreover, $(1/n)\sum_{k=1}^{n}X_k$ converges a.s. and in $L^1$ whenever $(X_n)_{n\geq 1}$ is (real-valued) c.i.d. and $E[\abs{X_1}]<\infty$. As to the $CLT$, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is $E[X_{n+1}\mid\mathcal{G}_n]$. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.