This paper surveys selected applications of the Lorenz curve and related stochastic orders in economics and econometrics, with a bias towards problems in statistical distribution theory. These include characterizations of income distributions in terms of families of inequality measures, Lorenz ordering of multiparameter distributions in terms of their parameters, probability inequalities for di...

We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integral...

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

Physical scenarios, leading to highly energetic stochastic gravitational waves backgrounds (for frequencies ranging from the μHz up to the GHz) are examined. In some cases the typical amplitude of the logarithmic energy spectrum can be even eight orders of magnitude larger than the ordinary inflationary prediction. Scaling violations in the frequency dependence of the energy density of the prod...

We consider a cointegrated system generated by processes that may be fractionally integrated, and by additive polynomial or generalized polynomial trends. In view of the consequent competition between stochastic and deterministic trends, we consider various estimates of the cointegrating vector and develop relevant asymptotic theory, including the situation where fractional orders of integratio...

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

For any given order of stochastic dominance, standard concentration curves are decomposed into contribution curves corresponding to within-group inequalities, between-group inequalities, and transvariational inequalities. We prove, for all orders, that contribution curve dominance implies systematically welfare-improving tax reforms and conversely. Accordingly, we point out some undesirable fis...

Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex coneH of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are eq...

The method of characteristics (the averaging over the characteristic formula) and the weak-sense numerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used to propose numerical methods for linear stochastic partial differential equations (SPDEs). Their orders of convergence in the mean-square sense and in the sense of almost sure converg...

The Cox process is a stochastic process which generalises the Poisson process by letting the underlying intensity function itself be a stochastic process. In this paper we present a fast Bayesian inference scheme for the permanental process, a Cox process under which the square root of the intensity is a Gaussian process. In particular we exploit connections with reproducing kernel Hilbert spac...