Open Access
Translator Disclaimer
July 2004 Limit theorems for a class of identically distributed random variables
Patrizia Berti, Luca Pratelli, Pietro Rigo
Ann. Probab. 32(3): 2029-2052 (July 2004). DOI: 10.1214/009117904000000676


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.


Download Citation

Patrizia Berti. Luca Pratelli. Pietro Rigo. "Limit theorems for a class of identically distributed random variables." Ann. Probab. 32 (3) 2029 - 2052, July 2004.


Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1050.60004
MathSciNet: MR2073184
Digital Object Identifier: 10.1214/009117904000000676

Primary: 60B10 , 60G09
Secondary: 60F05 , 60F15

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

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • July 2004
Back to Top