IMT Institutional Repository: No conditions. Results ordered -Date Deposited. 2024-10-09T13:53:54ZEPrintshttp://eprints.imtlucca.it/images/logowhite.pnghttp://eprints.imtlucca.it/2016-02-24T12:03:56Z2016-12-19T10:00:37Zhttp://eprints.imtlucca.it/id/eprint/3113This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/31132016-02-24T12:03:56ZAsymptotics for randomly reinforced urns with random barriersAn urn contains black and red balls. Let Zn be the proportion of black balls at time n and 0≤L<U≤1 random barriers. At each time n, a ball bn is drawn. If bn is black and Zn-1<U, then bn is replaced together with a random number Bn of black balls. If bn is red and Zn-1>L, then bn is replaced together with a random number Rn of red balls. Otherwise, no additional balls are added, and bn alone is replaced. In this paper we assume that Rn=Bn. Then, under mild conditions, it is shown that Zn→a.s.Z for some random variable Z, and Dn≔√n(Zn-Z)→Patrizia BertiIrene Crimaldiirene.crimaldi@imtlucca.itLuca PratelliPietro Rigo2015-02-23T08:34:38Z2015-02-23T08:34:38Zhttp://eprints.imtlucca.it/id/eprint/2612This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/26122015-02-23T08:34:38ZCentral Limit Theorems for an Indian Buffet Model with Random WeightsThe 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 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.Patrizia BertiIrene Crimaldiirene.crimaldi@imtlucca.itLuca PratelliPietro Rigo2013-06-21T11:23:24Z2014-01-29T14:34:19Zhttp://eprints.imtlucca.it/id/eprint/1622This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/16222013-06-21T11:23:24ZAn Anscombe-type theoremLet (X_n) be a sequence of random variables (with values in a
separable metric space) and (N_n) a sequence of random indices. Conditions for X_{N_n} to converge stably (in particular, in distribution) are provided. Some examples, where such conditions work but those already existing fail, are given as well.Patrizia BertiIrene Crimaldiirene.crimaldi@imtlucca.itLuca PratelliPietro Rigo2011-10-31T13:28:39Z2011-11-03T13:19:36Zhttp://eprints.imtlucca.it/id/eprint/980This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/9802011-10-31T13:28:39ZRate of convergence of predictive distributions for dependent dataThis 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},\ldots,X_{n})\},\]
where (Xn) is a sequence of random variables and μn=(1/n)∑i=1nδXi the empirical measure. Conditions for supB|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 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 $\sup_{B}|C_{n}(B)|\stackrel{P}{\rightarrow}0$ or even that $\sqrt{n}\sup_{B}|C_{n}(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.
Patrizia BertiIrene Crimaldiirene.crimaldi@imtlucca.itLuca PratelliPietro Rigo2011-10-31T12:04:05Z2011-11-03T13:19:36Zhttp://eprints.imtlucca.it/id/eprint/978This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/9782011-10-31T12:04:05ZCentral limit theorems for multicolor urns with dominated colorsAn 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. Let A n = diag (An,1,…,An,d) be the n-th reinforce matrix. Assuming that EAn,j=EAn,1 for all n and j, a few central limit theorems (CLTs) are available for such urns. In real problems, however, it is more reasonable to assume that EA n,j = EA n,1 whenever n ≥ 1 and 1 ≤ j ≤ d0 , liminfn EAn,1 > limsupn EAn,j whenever j > d0 for some integer 1≤d0≤d. Under this condition, the usual weak limit theorems may fail, but it is still possible to prove the CLTs for some slightly different random quantities. These random quantities are obtained by neglecting dominated colors, i.e., colors from d0+1 to d, and they allow the same inference on the urn structure. The sequence (An : n ≥ 1) is independent but need not be identically distributed. Some statistical applications are given as well.Patrizia BertiIrene Crimaldiirene.crimaldi@imtlucca.itLuca PratelliPietro Rigo2011-10-31T11:23:14Z2011-11-17T14:44:57Zhttp://eprints.imtlucca.it/id/eprint/975This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/9752011-10-31T11:23:14ZA central limit theorem and its applications to multicolor randomly reinforced urnsLet Xn be a sequence of integrable real random variables, adapted to a filtration (Gn). Define Cn = √{(1 / n)∑k=1nXk - E(Xn+1 | Gn)} and Dn = √n{E(Xn+1 | Gn) - Z}, where Z is the almost-sure limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn, Dn) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑k=1nX_k - Z} = Cn + Dn → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns. Patrizia BertiIrene Crimaldiirene.crimaldi@imtlucca.itLuca PratelliPietro Rigo