Limit theorems for a class of identically distributed random variables
| dc.creator | Berti, Patrizia | |
| dc.creator | Pratelli, Luca | |
| dc.creator | Rigo, Pietro | |
| dc.date | 2004-10-05 | |
| dc.date.accessioned | 2026-07-07T05:12:53Z | |
| dc.date.available | 2026-07-07T05:12:53Z | |
| dc.description | A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_{n\geq 1} is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\geq 0}, if it is adapted to (G_n)_{n\geq 0} and, for each n\geq 0, (X_k)_{k>n} is identically distributed given the past G_n. In case G_0={\varnothing,Ω} and G_n=σ(X_1,...,X_n), a result of Kallenberg implies that (X_n)_{n\geq 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 (X_n)_{n\geq 1} is exchangeable if and only if (X_{τ(n)})_{n\geq 1} is c.i.d. for any finite permutation τof {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)\sum_{k=1}^nX_k converges a.s. and in L^1 whenever (X_n)_{n\geq 1} is (real-valued) c.i.d. and E[| X_1| ]<\infty. 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}| G_n]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/math/0410105 | |
| dc.identifier | http://arxiv.org/abs/math/0410105 | |
| dc.identifier | Annals of Probability 2004, Vol. 32, No. 3A, 2029-2052 | |
| dc.identifier | doi:10.1214/009117904000000676 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72748 | |
| dc.subject | Probability | |
| dc.subject | 60B10, 60G09 (Primary) 60F05, 60F15. (Secondary) | |
| dc.title | Limit theorems for a class of identically distributed random variables | |
| dc.type | text |