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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Abstract

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

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

Dates
First available in Project Euclid: 14 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1089808418

Digital Object Identifier
doi:10.1214/009117904000000676

Mathematical Reviews number (MathSciNet)
MR2073184

Zentralblatt MATH identifier
1050.60004

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1089808418


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