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,\Omega} and G_n=\sigma(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_{\tau(n)})_{n\geq 1} is c.i.d. for any finite permutation \tau of {1,2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-\sigma-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.Comment: 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/10.1214/00911790400000067

Let (X n ) be a sequence of integrable real random variables, adapted to a filtration (G n ).stably are given, where U and V are certain random variables. In particular, under such conditions, we obtainstably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

No abstract

No abstract

Let (Ω, B, P ) be a probability space, A ⊂ B a sub-σ-field, and µ a regular conditional distribution for P given A. Necessary and sufficient conditions for µ(ω)(A) to be 0-1, for all A ∈ A and ω ∈ A0, where A0 ∈ A and P (A0) = 1, are given. Such conditions apply, in particular, when A is a tail sub-σ-field. Let H(ω) denote the Aatom including the point ω ∈ Ω. Necessary and sufficient conditions for µ(ω)(H(ω)) to be 0-1, for all ω ∈ A0, are also given. If (Ω, B) is a standard space, the latter 0-1 law is true for various classically interesting sub-σ-fields A, including tail, symmetric, invariant, as well as some sub-σ-fields connected with continuous time processes.

No abstract

The three-parameter Indian buffet process is generalized. The possibly different role played by customers is taken into account by suitable (random) weights. Various limit theorems are also proved for such generalized Indian buffet process. Let Ln be the number of dishes experimented by the first n customers, and let Kn = (1/n)Ki where Ki is the number of dishes tried by customer i. The asymptotic distributions of Ln and Kn, suitably centered and scaled, are obtained. The convergence turns out to be stable (and not only in distribution). As a particular case, the results apply to the standard (i.e., nongeneralized) Indian buffet process.

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{\mu_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and $\mu_n=(1/n)\sum_{i=1}^n\delta_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ191 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm