Showing papers by "Luca Pratelli published in 2015"
TL;DR: In this article, the authors generalized the three-parameter Indian buffet process and proved that the convergence is stable and not only in distribution, but also in the case of non-generalized Indian buffets.
Abstract: The three-parameter Indian buffet process is generalized. T
he 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 L_n be the number of dishes experimented by the first n customers, and let
{\bar K}_n=(1/n)\sum_{i=1}^n K_i
where K_i is the number of dishes tried by customer i. The asymptotic distributions of L_n and {\bar K}_n, 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., non generalized) Indian buffet process.
12 citations
TL;DR: In this article, the existence of a probability P on a convex cone of real random variables on the probability space (,A,P0) such that PP0, EP|X| < 1 and EP(X) � 0 for all X 2 L is investigated.
Abstract: Let L be a convex cone of real random variables on the probability space (,A,P0). The existence of a probability P on A such that PP0, EP|X| < 1 and EP(X) � 0 for all X 2 L is investigated. Two results are provided. In the first, P is a finitely additive probability, while P is �-additive in the second. If L is a linear space then X 2 L whenever X 2 L, so that EP(X) � 0 turns into EP(X) = 0. Hence, the results apply to various significant frameworks, including equivalent mar- tingale measures and equivalent probability measures with given marginals.
11 citations
TL;DR: In this paper, the existence of such a martingale measure and its conditions for the existence are discussed. But they do not specify the conditions for such a measure to be useful in mass transportation.
Abstract: Let(X,A)and(Y,B)bemeasurablespaces.Supposewearegivenaprobability onA,aprobability onBandaprobabilityµ ontheproduct -eld A B.Isthereaprobability onA B,withmarginals and , such that µ or µ ? Such a , provided it exists, may be useful with regard to equivalent martingale measures and mass transportation. Various conditions for the existence of are provided, distinguishing µ from µ.
6 citations
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TL;DR: In this paper, it was shown that under mild conditions, the conditional convergence of the conditional distribution of a random variable Z = B_n with respect to the proportion of black and red balls in an urn is almost sure.
Abstract: An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\overset{a.s.}\longrightarrow Z$ for some random variable $Z$, and \begin{gather*} D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,\sigma^2)\quad\text{conditionally a.s.} \end{gather*} where $\sigma^2$ is a certain random variance. Almost sure conditional convergence means that \begin{gather*} P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)\overset{weakly}\longrightarrow\mathcal{N}(0,\,\sigma^2)\quad\text{a.s.} \end{gather*} where $P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)$ is a regular version of the conditional distribution of $D_n$ given the past $\mathcal{G}_n$. Thus, in particular, one obtains $D_n\longrightarrow\mathcal{N}(0,\sigma^2)$ stably. It is also shown that $L