arXiv subjects

Eugenio Regazzini

Publications and source records attributed to Eugenio Regazzini.

16 recordsLinked to original sources

arXiv2025

On the number of elements beyond the ones actually observed

In this work, a variant of the birth and death chain with constant intensities, originally introduced by Bruno de Finetti way back in 1957, is revisited. This fact is also underlined by the choice of the title, which is clearly a literal translation of the original one. Characteristic of the variant is that it allows negative jumps of any magnitude. And this, as explained in the paper, might be useful in offering some insight into the issue, arising in numerous situations, of inferring the number of the undetected elements of a given population. One thinks, for example, of problems concerning abundance or richness of species. The author's purpose is twofold: to align the original de Finetti's construction with the modern, well-established theory of the continuous-time Markov chains with discrete state space and show how it could be used to make probabilistic previsions on the number of the unseen elements of a population. With the aim of enhancing the possible practical applications of the model, one discusses the statistical point estimation of the rates which characterize its infinitesimal description.

math.PR↗

arXiv2018

Uniform rates of the Glivenko-Cantelli convergence and their use in approximating Bayesian inferences

This paper deals with the problem of quantifying the approximation a probability measure by means of an empirical (in a wide sense) random probability measure, depending on the first n terms of a sequence of random elements. In Section 2, one studies the range of oscillation near zero of the Wasserstein distance $\ud^{(p)}_{\pms}$ between $\pfrak_0$ and $\hat{\pfrak}_n$, assuming that the $\xitil_i$'s are i.i.d. with $\pfrak_0$ as common law. Theorem 2.3 deals with the case in which $\pfrak_0$ is fixed as a generic element of the space of all probability measures on $(\rd, \mathscr{B}(\rd))$ and $\hat{\pfrak}_n$ coincides with the empirical measure. In Theorem 2.4 (Theorem 2.5, respectively) \pfrak_0 is a d-dimensional Gaussian distribution (an element of a distinguished type of statistical exponential family, respectively) and $\hat{\pfrak}_n$ is another $d$-dimensional Gaussian distribution with estimated mean and covariance matrix (another element of the same family with an estimated parameter, respectively). These new results improve on allied recent works (see, e.g., [31]) since they also provide uniform bounds with respect to $n$, meaning that the finiteness of the p-moment of the random variable $\sup_{n \geq 1} b_n \ud^{(p)}_{\pms}(\pfrak_0, \hat{\pfrak}_n)$ is proved for some suitable diverging sequence b_n of positive numbers. In Section 3, under the hypothesis that the $\xitil_i$'s are exchangeable, one studies the range of the random oscillation near zero of the Wasserstein distance between the conditional distribution--also called posterior--of the directing measure of the sequence, given $\xitil_1, \dots, \xitil_n$, and the point mass at $\hat{\pfrak}_n$. In a similar vein, a bound for the approximation of predictive distributions is given. Finally, Theorems from 3.3 to 3.5 reconsider Theorems from 2.3 to 2.5, respectively, according to a Bayesian perspective.

math.PR↗

arXiv2016

Inequality and risk aversion in economies open to altruistic attitudes

This paper attempts to find a relationship between agents' risk aversion and inequality of incomes. Specifically, a model is proposed for the evolution in time of surplus/deficit distribution, and the long-time distributions are characterized almost completely. They turn out to be weak Pareto laws with exponent linked to the relative risk aversion index which, in turn, is supposed to be the same for every agent. On the one hand, the aforesaid link is expressed by an affine transformation. On the other hand, the level of the relative risk aversion index results from a frequency distribution of observable quantities stemming from how agents interact in an economic sense. Combination of these facts is conducive to the specification of qualitative and quantitative characteristics of actions fit for the control of income concentration.

econ.GN↗

arXiv2016

Frequentistic approximations to Bayesian prevision of exchangeable random elements

Given a sequence ξ_1, ξ_2,... of X-valued, exchangeable random elements, let q(ξ^(n)) and p_m(ξ^(n)) stand for posterior and predictive distribution, respectively, given ξ^(n) = (ξ_1,..., ξ_n). We provide an upper bound for limsup b_n d_[[X]](q(ξ^(n)), δ_\empiricn) and limsup b_n d_[X^m](p_m(ξ^(n)), \empiricn^m), where \empiricn is the empirical measure, b_n is a suitable sequence of positive numbers increasing to +\infty, d_[[X]] and d_[X^m] denote distinguished weak probability distances on [[X]] and [X^m], respectively, with the proviso that [S] denotes the space of all probability measures on S. A characteristic feature of our work is that the aforesaid bounds are established under the law of the ξ_n's, unlike the more common literature on Bayesian consistency, where they are studied with respect to product measures (p_0)^\infty, as p_0 varies among the admissible determinations of a random probability measure.

math.ST↗

arXiv2015

Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations

For a general inelastic Kac-like equation recently proposed, this paper studies the long-time behaviour of its probability-valued solution. In particular, the paper provides necessary and sufficient conditions for the initial datum in order that the corresponding solution converges to equilibrium. The proofs rest on the general CLT for independent summands applied to a suitable Skorokhod representation of the original solution evaluated at an increasing and divergent sequence of times. It turns out that, roughly speaking, the initial datum must belong to the standard domain of attraction of a stable law, while the equilibrium is presentable as a mixture of stable laws.

math.PR↗

arXiv2014

Probabilistic View of Explosion in an Inelastic Kac Model

Let $\{μ(\cdot,t):t\geq0\}$ be the family of probability measures corresponding to the solution of the inelastic Kac model introduced in Pulvirenti and Toscani [\textit{J. Stat. Phys.} \textbf{114} (2004) 1453-1480]. It has been proved by Gabetta and Regazzini [\textit{J. Statist. Phys.} \textbf{147} (2012) 1007-1019] that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law. In the present paper it is shown that, for initial data which are heavier-tailed than the aforementioned ones, the limiting distribution is improper in the sense that it has probability 1/2 "adherent" to $-\infty$ and probability 1/2 "adherent" to $+\infty$. It is explained in which sense this phenomenon is amenable to a sort of explosion, and the main result consists in an explicit expression of the rate of such an explosion. The presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved. This gives the chance to make an adjustment to a portion of a proof contained in the above-mentioned paper by Gabetta and Regazzini.

math.PR↗

arXiv2012

Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions

The present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules. The beginning of the story dates back to a pioneering work by Hilbert, who first formalized the concept of linearization of the collision operator and pointed out the importance of its eigenvalues with respect to a certain asymptotic behavior of the Boltzmann equation. Under really mild conditions on initial data - close to being necessary - and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 1.1) contains the first precise statement that the total variation distance between the solution and the limiting Maxwellian distribution admits an upper bound of the form C e^{Lambda_b t}, Lambda_b being the least negative of the aforesaid eigenvalues and C a constant which depends only on a few simple numerical characteristics (e.g. moments) of the initial datum. The validity of this quantification was conjectured, about fifty years ago, in a paper by Henry P. McKean but, in spite of several attempts, the best answer known up to now consists in a bound with a rate which can be made arbitrarily close to Lambda_b, to the cost of the "explosion" of the constant C. Moreover, its deduction is subject to restrictive hypotheses on the initial datum, besides the Grad angular cutoff condition. As to the proof of our results, we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, highlighted by McKean. Our work represents in fact a confirmation of this analogy, since the techniques we develop here crucially rely on certain formulations of the central limit theorem.

math-ph↗

arXiv2012

The origins of de Finetti's critique of countable additivity

Bruno de Finetti was one of the most convinced advocates of finitely additive probabilities. The present work describes the intellectual pro- cess that led him to support that stance and provides a detailed account both of the first paper by de Finetti on the subject and of the ensuing correspondence with Maurice Fréchet. Moreover, the analysis is supplemented by a useful picture of de Finetti's interactions with the international scientific community at that time, when he elaborated his subjectivistic conception of probability.

math.ST↗

arXiv2012

Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits

The problem of convergence in law of normed sums of exchangeable random variables is examined. First, the problem is studied w.r.t. arrays of exchangeable random variables, and the special role played by mixtures of products of stable laws - as limits in law of normed sums in different rows of the array - is emphasized. Necessary and sufficient conditions for convergence to a specific form in the above class of measures are then given. Moreover, sufficient conditions for convergence of sums in a single row are proved. Finally, a potentially useful variant of the formulation of the results just summarized is briefly sketched, a more complete study of it being deferred to a future work.

math.PR↗

arXiv2012

Complete characterization of convergence to equilibrium for an inelastic Kac model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index $α$ is determined by the so-called degree of inelasticity, $p>0$, of the particles: $α=\frac{2}{1+p}$. This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index $α$. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.

math.PR↗

arXiv2011

Probabilistic representation for the solution of the homogeneous Boltzmann equation for Maxwellian molecules

Consider the homogeneous Boltzmann equation for Maxwellian molecules. We provide a new representation for its solution in the form of expectation of a random probability measure M. We also prove that the Fourier transform of M is a conditional characteristic function of a sum of independent random variables, given a suitable sigma-algebra. These facts are then used to prove a CLT for Maxwellian molecules, that is the statement of a necessary and sufficient condition for the weak convergence of the solution of the equation. Such a condition reduces to the finiteness of the second moment of the initial distribution μ_0. As a further application, we give a refinement of some inequalities, due to Elmroth, concerning the evolution of the moments of the solution.

math.PR↗

arXiv2010

The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation

In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186-201] it is proved that the total variation distance between the solution $f(\cdot,t)$ of Kac's equation and the Gaussian density $(0,σ^2)$ has an upper bound which goes to zero with an exponential rate equal to -1/4 as $t\to+\infty$. In the present paper, we determine a lower bound which decreases exponentially to zero with this same rate, provided that a suitable symmetrized form of $f_0$ has nonzero fourth cumulant $κ_4$. Moreover, we show that upper bounds like $\bar{C}_δe^{-({1/4})t}ρ_δ(t)$ are valid for some $ρ_δ$ vanishing at infinity when $\int_{\mathbb{R}}|v|^{4+δ}f_0(v)\,dv<+\infty$ for some $δ$ in $[0,2[$ and $κ_4=0$. Generalizations of this statement are presented, together with some remarks about non-Gaussian initial conditions which yield the insuperable barrier of -1 for the rate of convergence.

math.PR↗

arXiv2009

Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem

Let $f(\cdot,t)$ be the probability density function which represents the solution of Kac's equation at time $t$, with initial data $f_0$, and let $g_σ$ be the Gaussian density with zero mean and variance $σ^2$, $σ^2$ being the value of the second moment of $f_0$. This is the first study which proves that the total variation distance between $f(\cdot,t)$ and $g_σ$ goes to zero, as $t\to +\infty$, with an exponential rate equal to -1/4. In the present paper, this fact is proved on the sole assumption that $f_0$ has finite fourth moment and its Fourier transform $φ_0$ satisfies $|φ_0(ξ)|=o(|ξ|^{-p})$ as $|ξ|\to+\infty$, for some $p>0$. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.

math.PR↗

arXiv2009

Central limit theorem for the solution of the Kac equation

We prove that the solution of the Kac analogue of Boltzmann's equation can be viewed as a probability distribution of a sum of a random number of random variables. This fact allows us to study convergence to equilibrium by means of a few classical statements pertaining to the central limit theorem. In particular, a new proof of the convergence to the Maxwellian distribution is provided, with a rate information both under the sole hypothesis that the initial energy is finite and under the additional condition that the initial distribution has finite moment of order $2+δ$ for some $δ$ in $(0,1]$. Moreover, it is proved that finiteness of initial energy is necessary in order that the solution of Kac's equation can converge weakly. While this statement may seem to be intuitively clear, to our knowledge there is no proof of it as yet.

math.PR↗

arXiv2008

Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model

This paper deals with a one--dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter $p>0$. In particular, the paper provides bounds for certain distances -- such as specific weighted $χ$--distances and the Kolmogorov distance -- between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent $\a=2/(1+p)$. With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra--condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, near to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function.

math-ph↗

arXiv2004

Means of a Dirichlet process and multiple hypergeometric functions

The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these are a new and more direct procedure for determining the exact form of the distribution of the mean, a correspondence between the distribution of the mean and the parameter of a Dirichlet process, a characterization of the family of Cauchy distributions as the set of the fixed points of this correspondence, and an extension of the Markov-Krein identity. Moreover, an expression of the characteristic function of the mean of a Dirichlet process is obtained by resorting to an integral representation of a confluent form of the fourth Lauricella function. This expression is then employed to prove that the distribution of the mean of a Dirichlet process is symmetric if and only if the parameter of the process is symmetric, and to provide a new expression of the moment generating function of the variance of a Dirichlet process.

math.PR↗