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Let (Ω, B) be a measurable space, An ⊂ B a sub-σ-field and µn a random probability measure on (Ω, B), n ≥ 1. In various frameworks, one looks for a probability P on B such that µn is a regular conditional distribution for P given An for all n. Conditions for such a P to exist are given. The conditions are quite simple when (Ω, B) is a compact Hausdorff… (More)

Let (µn : n ≥ 0) be Borel probabilities on a metric space S such that µn → µ 0 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 → X 0 in probability. By Skorohod's theorem, Skorohod representation holds (with Xn → X 0 almost uniformly) if µ 0 is… (More)

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (Xn) n≥1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (Gn) n≥0 , if it is adapted to (Gn) n≥0 and, for each n ≥ 0, (X k) k>n is identically distributed given the past Gn. In case G0 = {∅, Ω} and Gn =… (More)

Let L be a linear space of real bounded random variables on the probability space (Ω, A, P 0). There is a finitely additive probability P on A, such that P ∼ P 0 and E P (X) = 0 for all X ∈ L, if and only if c E Q (X) ≤ ess sup(−X), X ∈ L, for some constant c > 0 and (countably additive) probability Q on A such that Q ∼ P 0. A necessary condition for such a… (More)

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) n i=1 K i where K i is the… (More)

Let (Xn) be a sequence of integrable real random variables, adapted to a filtration (Gn). Define

This paper deals with empirical processes of the type where (Xn) is a sequence of random variables and µn = (1/n) n i=1 δ X i the empirical measure. Conditions for sup B |Cn(B)| to converge stably (in particular , in distribution) are given, where B ranges over a suitable class of measurable sets. These conditions apply when (Xn) is exchangeable, or, more… (More)

Let (Ω, A, P) be a probability space, S a metric space, µ a probability measure on the Borel σ-field of S, Xn → X a.s. (without need of taking subsequences) if µ{x} = 0 for all x, or if P (Xn = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to σ(A ∪ {H}) for some H ⊂ Ω with P *… (More)

- Patrizia Berti, Pietro Rigo
- Annals of Mathematics and Artificial Intelligence
- 2002

Existence of coherent extensions of coherent conditional probabilities is one of the major merits of de Finetti's theory of probability. However, coherent extensions which meet some special property, like σ-additivity or disintegrability, can fail to exist. An example is given where a coherent and σ-additive conditional probability cannot be extended… (More)

An urn contains balls of d ≥ 2 colors. At each time n ≥ 1, a ball is drawn and then replaced together with a random number of balls of the same color. ¡ be the n-th reinforce matrix. Assuming EA n,j = EA n,1 for all n and j, a few CLT's are available for such urns. In real problems, however, it is more reasonable to assume EA n,j = EA n,1 whenever n ≥ 1 and… (More)