2004/10/05 by Patrizia Berti, Luca Pratelli, Pietro Rigo · 6 citations
Mathematics · #math.PR #msc:60B10 #msc:60G09 #msc:60F05 #msc:60F15.
paper · pdf · doi:10.1214/009117904000000676
published as Annals of Probability 2004, Vol. 32, No. 3A, 2029-2052 · Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000676
arxiv created 2004/10/05 · arxiv updated 2009/12/01
A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (Xn)n≥ 1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (Gn)n≥ 0, if it is adapted to (Gn)n≥ 0 and, for each n≥ 0, (Xk)k>n is identically distributed given the past Gn. In case G0=\varnothing,Ω and Gn=σ(X1,...,Xn), a result of Kallenberg implies that (Xn)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 (Xn)n≥ 1 is exchangeable if and only if (Xτ(n))n≥ 1 is c.i.d. for any finite permutation τof 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=1nXk converges a.s. and in L1 whenever (Xn)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[Xn+1| Gn]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.