Limit theorems for a class of identically distributed random variables

Abstract

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_n≥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≥0, (X_k)_k>n is identically distributed given the past $\mathcal{G}_{n}$ . In case $\mathcal{G}_{0}=\{\varnothing,\Omega\}$ and $\mathcal{G}_{n}=\sigma(X_{1},\ldots,X_{n})$ , a result of Kallenberg implies that (X_n)_n≥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≥1 is exchangeable if and only if (X_τ(n))_n≥1 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, (1/n)∑_k=1ⁿX_k converges a.s. and in L¹ whenever (X_n)_n≥1 is (real-valued) c.i.d. and E[|X₁|]<∞. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is $E[X_{n+1}\vert \mathcal{G}_{n}]$. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.