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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)

- MARTINGALE MEASURES, PIETRO RIGO
- 2010

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)

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)

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 (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)

J o u r n a l o f P r o b a b i l i t y Electron. Abstract Let L be a convex cone of real random variables on the probability space (Ω, A, P0). The existence of a probability P on A such that P ∼ P0, EP |X| < ∞ and EP (X) ≤ 0 for all X ∈ L is investigated. Two types of results are provided, according to P is finitely additive or σ-additive. The main results… (More)

In various frameworks, to assess the joint distribution of a k

Empirical processes for non ergodic data are investigated under uniform distance. Some CLT's, both uniform and non uniform, are proved. In particular, conditions for the empirical process B n = √ n(µ n − b n) to converge in distribution are given, where µ n is the empirical measure and b n the arithmetic mean of the first n predictive measures. Such… (More)