Convergence results for conditional expectations

Let E,F be two Polish spaces and [Xn,Yn],[X,Y] random variables with values in E×F (not necessarily defined on the same probability space). We show some conditions which are sufficient in order to assure that, for each bounded continuous function f on E×F, the conditional expectation of f(Xn,Yn) given Yn converges in distribution to the conditional expectation of f(X,Y) given Y

Limit theorems for a class of identically distributed random variables

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

Atomic Intersection of s-Fields and Some of Its Consequences

Let (omega,F,P) be a probability space. For each G in F, define G as the s-field generated by G and those sets f in F satisfying P(f) in {0, 1}. Conditions for P to be atomic on the intersection of the complements of Ai for i=1,..,k, with A1, . . . ,Ak in F sub-s-fields, are given. Conditions for P to be 0-1-valued on the intersection of the complements of Ai for i=1,..,k are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.Atomic probability measure, Gibbs sampling, Graphical models, Intersection property, Iterated conditional expectations.

Skorohod Representation Theorem Via Disintegrations

Let (µn : n >= 0) be Borel probabilities on a metric space S such that µn -> µ0 weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables Xn satisfying Xn - µn for all n and Xn -> X0 in probability. By Skorohod’s theorem, Skorohod representation holds (with Xn -> X0 almost uniformly) if µ0 is separable. Two results are proved in this paper. First, Skorohod representation may fail if µ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of µ0 separable or not, Skorohod representation holds if W(µn, µ0) -> 0 where W is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W), disintegrable probability measures are fundamental.Disintegration, Separable probability measure, Skorohod representation theorem, Wasserstein distance, Weak convergence of probability measures.

Limit Theorems for Empirical Processes Based on Dependent Data

Empirical processes for non ergodic data are investigated under uniform distance. Some CLTs, both uniform and non uniform, are proved. In particular, conditions for Bn = n^(1/2) (µn - bn) and Cn = n^(1/2) (µn - an) to converge in distribution are given, where µn is the empirical measure, an the predictive measure, and bn = 1/n sum (ai) for i=0 to n-1. Such conditions can be applied to any adapted sequence of random variables. Various examples and a characterization of conditionally identically distributed sequences are given as well.Conditional identity in distribution, empirical process, exchangeability, predictive measure, stable convergence.

Finitely Additive Equivalent Martingale Measures

Let L be a linear space of real bounded random variables on the probability space (omega,A, P0). There is a finitely additive probability P on A, such that P tilde P0 and EP (X) = 0 for all X in L, if and only if cEQ(X) = ess sup(-X), X in L, for some constant c > 0 and (countably additive) probability Q on A such that Q tilde P0. A necessary condition for such a P to exist is L - L+(inf) n L+(inf) = {0}, where the closure is in the norm-topology. If P0 is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability P on A, such that PArbitrage, de Finetti’s coherence principle, equivalent martingale measure, finitely additive probability, fundamental theorem of asset pricing.

Exchangeable Sequences Driven by an Absolutely Continuous Random Measure

Let S be a Polish space and (Xn : n = 1) an exchangeable sequence of S-valued random variables. Let an(·) = P( Xn+1 in · | X1, . . . ,Xn) be the predictive measure and a a random probability measure on S such that an (weak) --> a a.s.. Two (related) problems are addressed. One is to give conditions for a 0, where ||·|| is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that (Xn) is conditionally identically distributed, in the sense of [2].Conditional identity in distribution, Exchangeability, Predictive measure, Random probability measure.

A Skorohod Representation Theorem for Uniform Distance

Let µn be a probability measure on the Borel sigma-field on D[0, 1] with respect to Skorohod distance, n = 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables Xn such that Xn tilde µn for all n = 0 and ||Xn - X0|| --> 0 in probability, where ||·|| is the sup-norm. Such conditions do not require µ0 separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.Cadlag function – Exchangeable empirical process – Separable probability measure – Skorohod representation theorem– Uniform distance – Weak convergence of probability measures.

Rate of convergence of predictive distributions for dependent data

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