Rate of convergence of predictive distributions for dependent data

This paper deals with empirical processes of the type $$C_n(B)=\sqrt{n}\{\mu_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},$$ where $(X_n)$ is a sequence of random variables and $\mu_n=(1/n)\sum_{i=1}^n\delta_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ 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 $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ191 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotics for randomly reinforced urns with random barriers

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L<U\leq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, 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 $L<Z<U$ a.s. and $Z$ has non-atomic distribution.Comment: 13 pages, submitte

Rate of Convergence of Predictive Distributions for Dependent Data

This paper deals with empirical processes of the type Cn(B) = n^(1/2) {µn(B) - P(Xn+1 in B | X1, . . . ,Xn)} , where (Xn) is a sequence of random variables and µn = (1/n)SUM(i=1,..,n) d(Xi) 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 [6]). By such conditions, in some relevant situations, one obtains that supB|Cn(B)|-P->0 or even that n^(1/2) supB|Cn(B)| converges a.s.. Results of this type are useful in Bayesian statistics.Bayesian predictive inference, Central limit theorem, Conditional identity in distribution, Empirical distribution, Exchangeability, Predictive distribution, Stable convergence.

Central Limit Theorems For Multicolor Urns With Dominated Colors

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. Let An =diag (An,1, . . . ,An,d) be the n-th reinforce matrix. Assuming EAn,j = EAn,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 EAn,j = EAn,1 whenever n >= 1 and 1 limsup EAn,j whenever j > d0, for some integer 1 = 1) is independent but need not be identically distributed. Some statistical applications are given as well.Central limit theorem, Clinical trials, Random probability measure, Stable convergence, Urn model.

Central limit theorems for multicolor urns with dominated colors

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. 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

Legius syndrome: Case report and review of literature

A 8-month-old child was referred to our Dermatologic Unit for suspected Neurofibromatosis type 1 (NF 1), because of the appearance, since few days after birth, of numerous caf\ue9-au-lait spots (seven larger than 5 mm); no other sign evocative of NF 1 was found. Her family history was remarkable for the presence of multiple caf\ue9-au-lait spots in the mother, the grandfather and two aunts. The family had been already examined for NF 1, but no sign evocative of the disease was found. We then suspected Legius syndrome, a dominant disease characterized by a mild neurofibromatosis 1 phenotype. The diagnosis was confirmed by the finding of a mutation in SPRED1 gene, a feedback regulator of RAS/MAPK signaling. Here, we discuss the differential diagnosis of caf\ue8-au-lait spots and we briefly review the existing literature about Legius syndrome

Quanto costa incidere una stele? Costi di produzione e meccanismi di pubblicazione delle iscrizioni pubbliche in Grecia

Il contributo affronta la questione del finanziamento e del costo dei decreti pubblici e si propone di ricostruire, sulla base dell’ analisi dei casi esemplari di Atene, Delo e Cos, i fattori economici che determinavano la formazione del prezzo di una stele pubblica. A questo scopo si analizzano i processi di pubblicazione e produzione, dallo stanziamento dei fondi e dal bando dei contratti, al reperimento di materiale e mano d’opera, all’organizzazione della bottega, fino al trasporto e alla collocazione del prodotto finito.  This paper analyzes the production of a marble stele and its costs from the public decision to publish an inscription to the erection of the final product. On the basis of the case studies of Athens, Cos and Delos, the paper discusses the supply of materials, the work of the mason, the labor cost and the letting out of the contracts for the production

Lisbeth Bredholt Christensen et Jesper Tae Jensen (éds.), Religion and Material Culture: Studying Religion and Religious Elements on the Basis of Objects, Architecture and Space

Il volume raccoglie gli atti di una conferenza tenutasi nel 2011 al Centre for Bible and Cultural Memory di Copenhagen e al National Museum of Denmark. La novità della pubblicazione è nell’affrontare attraverso un approccio materiale non solo il tema del culto (su cui già esiste vastissima letteratura), ma più in generale della religione, intesa anche nei suoi aspetti teologici, astratti e discorsivi. Intenzionalmente, la raccolta non si limita ad analizzare un solo ambito culturale o cronolo..