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

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 ^{(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 ^{(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

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 $(0,\sigma^2)$ 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 $\kappa_4$. Moreover, we show that upper bounds like $\bar{C}_{\delta}e^{-({1/4})t}\rho_{\delta}(t)$ are valid for some $\rho_{\delta}$ vanishing at infinity when $\int_{\mathbb{R}}|v|^{4+\delta}f_0(v)\,dv<+\infty$ for some $\delta$ in $[0,2[$ and $\kappa_4=0$. 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.Comment: Published in at http://dx.doi.org/10.1214/09-AAP623 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Probabilistic View of Explosion in an Inelastic Kac Model

Let $\{\mu(\cdot,t):t\geq0\}$ 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

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

Frequentistic approximations to Bayesian prevision of exchangeable random elements

Given a sequence \xi_1, \xi_2,... of X-valued, exchangeable random elements, let q(\xi^(n)) and p_m(\xi^(n)) stand for posterior and predictive distribution, respectively, given \xi^(n) = (\xi_1,..., \xi_n). We provide an upper bound for limsup b_n d_[[X]](q(\xi^(n)), \delta_\empiricn) and limsup b_n d_[X^m](p_m(\xi^(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 \xi_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

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.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000027

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 $2+\delta$ for some $\delta$ 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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP524 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org