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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, (Xk)k>n is identically distributed given the past Gn. In case G0 = {∅,Ω} and Gn = σ(X1, . . .… (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 Ki where Ki is the… (More)

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 space… (More)

Let (Ω,A, P ) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and Xn : Ω → S an arbitrary map, n = 1, 2, . . .. If μ is tight and Xn converges in distribution to μ (in HoffmannJørgensen’s sense), then X ∼ μ for some S-valued random variable X on (Ω,A, P ). If, in addition, the Xn are measurable and tight, there… (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)

Let (Ω,F , P ) be a probability space and N the class of those F ∈ F satisfying P (F ) ∈ {0, 1}. For each G ⊂ F , define G = σ G ∪ N . Necessary and sufficient conditions for A∩B = A ∩ B, where A,B ⊂ F are subσ-fields, are given. These conditions are then applied to the (two component) Gibbs sampler. Suppose X and Y are the coordinate projections on (Ω,F) =… (More)

- Pietro Rigo
- 2006

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.… (More)

Let (S, d) be a metric space, G a σ-field on S and (μn : n ≥ 0) a sequence of probabilities on G. Suppose G countably generated, the map (x, y) 7→ d(x, y) measurable with respect to G ⊗ G, and μn perfect for n > 0. Say that (μn) has a Skorohod representation if, on some probability space, there are random variables Xn such that Xn ∼ μn for all n ≥ 0 and… (More)

Let L be a linear space of real bounded random variables on the probability space (Ω,A, P0). There is a finitely additive probability P on A, such that P ∼ P0 and EP (X) = 0 for all X ∈ L, if and only if c EQ(X) ≤ ess sup(−X), X ∈ L, for some constant c > 0 and (countably additive) probability Q on A such that Q ∼ P0. A necessary condition for such a P to… (More)

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