Limit theorems for a class of identically distributed random variables

2004/07/01 by Patrizia Berti, Luca Pratelli, Pietro Rigo · 11 citations
Computer Science · Decision Sciences · Mathematics · #Bayesian Methods and Mixture Models #Probability and Risk Models #Statistical Methods and Inference #math.PR #msc:60B10 #msc:60F05 #msc:60F15. #msc:60G09

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

openalex publication_date 2004/07/01 · arxiv created 2004/10/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

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 (G_n)_{n≥ 0} , if it is adapted to (G_n)_{n≥ 0} and, for each n≥0, (Xk)k>n is identically distributed given the past G_n . In case G₀=\\varnothing,Ω\ and G_n=σ(X₁,…,X_n) , 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 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=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[X_n+1\vert G_n]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.

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