Families of infinitely divisible distributions closed under mixing and convolution

Certain families of probability distribution functions maintain their infinite divisibility under repeated mixing and convolution. Examples on the continuum and lattice are given. The main tools used are Polya's criteria and the properties of log-convexity and complete monotonicity. Some light is shed on the relationship between these two properties

Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new section on mod-Gaussian convergence coming from the factorization of the generating function ; the multi-dimensional results have been moved to a forthcoming paper ; and the introduction has been reworke

Renewal theory and level passage by subordinators

Limit theorems in renewal processes generated by infinitely divisible life times easily yield formulas for the limit distributions of the 'undershoot' and 'overshoot' at the passage of a level by subordinators

Asymptotic behaviour of estimators of the parameters of nearly unstable INAR(1) models

A sequence of first-order integer-valued autoregressive type (INAR(1)) processes is investigated, where the autoregressive type coefficients converge to 1. It is shown that the limiting distribution of the joint conditional least squares estimators for this coefficient and for the mean of the innovation is normal. Consequences for sequences of Galton{Watson branching processes with unobservable immigration, where the mean of the offspring distribution converges to 1 (which is the critical value), are discussed

Educating through Exemplars: Alternative Paths to Virtue

This paper confronts Zagzebski’s exemplarism with the intertwined debates over the conditions of exemplarity and the unity-disunity of the virtues, to show the advantages of a pluralistic exemplar-based approach to moral education (PEBAME). PEBAME is based on a prima facie disunitarist perspective in moral theory, which amounts to admitting both exemplarity in all respects and single-virtue exemplarity. First, we account for the advantages of PEBAME, and we show how two figures in recent Italian history (Giorgio Perlasca and Gino Bartali) satisfy Blum’s definitions of ‘moral hero’ and ‘moral saint’ (1988). Then, we offer a comparative analysis of the effectiveness of heroes and saints with respect to character education, according to four criteria derived from PEBAME: admirability, virtuousness, transparency, and imitability. Finally, we conclude that both unitarist and disunitarist exemplars are fundamental to character education; this is because of the hero's superiority to the saint with respect to imitability, a fundamental feature of the exemplar for character education

Modelling overdispersion with integer-valued moving average processes

A new first-order integer-valued moving average, INMA(1), model based on the negative binomial thinning operation defined by Risti´c et al. [21] is proposed and characterized. It is shown that this model has negative binomial (NB) marginal distribution when the innovations follow a NB distribution and therefore it can be used in situations where the data present overdispersion. Additionally, this model is extended to the bivariate context. The Generalized Method of Moments (GMM) is used to estimate the unknown parameters of the proposed models and the results of a simulation study that intends to investigate the performance of the method show that, in general, the estimates are consistent and symmetric. Finally, the proposed model is fitted to a real dataset and the quality of the adjustment is evaluated.publishe

Statistical modelling of counts with a simple integer-valued bilinear process

The aim of this work is the statistical modelling of counts assuming low values and exhibiting sudden and large bursts that occur randomly in time. It is well known that bilinear processes capture these kind of phenomena. In this work the integer-valued bilinear INBL(1,0,1,1) model is discussed and some properties are reviewed. Classical and Bayesian methodologies are considered and compared through simulation studies, namely to obtain estimates of model parameters and to calculate point and interval predictions. Finally, an empirical application to real epidemiological count data is also presented to attest for its practical applicability in data analysis.publishe

Citizenship Education and Liberalism: A State of the Debate Analysis 1990–2010

What kind of citizenship education, if any, should schools in liberal societies promote? And what ends is such education supposed to serve? Over the last decades a respectable body of literature has emerged to address these and related issues. In this state of the debate analysis we examine a sample of journal articles dealing with these very issues spanning a twenty-year period with the aim to analyse debate patterns and developments in the research field. We first carry out a qualitative analysis where we design a two-dimensional theoretical framework in order to systematise the various liberal debate positions, and make us able to study their justifications, internal tensions and engagements with other positions. In the ensuing quantitative leg of the study we carry out a quantitative bibliometric analysis where we weigh the importance of specific scholars. We finally discuss possible merits and flaws in the research field, as evidenced in and by the analysis