Limit theorems for a class of identically distributed random variables

By Patrizia Berti, Luca Pratelli, Pietro Rigo. 2004

104 citations recorded by the source corpus.

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 $(\mathcal{G}_{n})_{n\geq 0}$ , if it is adapted to $(\mathcal{G}_{n})_{n\geq 0}$ and, for each n≥0, (Xk)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 (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 $E[X_{n+1}\vert \mathcal{G}_{n}]$ . For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.

Subfields: Artificial Intelligence, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ba411c383a7f3924a5f443e59d9a636bcaa0f584

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Citations in the corpus

Martingale posterior distributions (2021, methodology): “Our choice of (1.1) will be density estimators based on recent ideas in the literature, specifically the conditionally identically distributed (c.i.d.) sequence of Berti et al. (2004) and bivariate copula update of Hahn et al. (2018).”
Almost sure weak convergence of random probability measures (2006, unknown): “[3] and [10].”