The Annals of Probability

Limit theorems for a class of identically distributed random variables

Patrizia Berti, Luca Pratelli, and Pietro Rigo

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A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (Xn)n1 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 n0, (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)n1 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)n1 is exchangeable if and only if (Xτ(n))n1 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)n1 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.

Article information

Ann. Probab., Volume 32, Number 3 (2004), 2029-2052.

First available in Project Euclid: 14 July 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures 60G09: Exchangeability
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems

Central limit theorem convergence [almost sure, in distribution, σ(L^{1},L^{∞}), stable] empirical process exchangeability strong law of large numbers uniform limit theorem


Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Limit theorems for a class of identically distributed random variables. Ann. Probab. 32 (2004), no. 3, 2029--2052. doi:10.1214/009117904000000676.

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