Victor Pérez-Abreu

Chaos expansions of double intersection local time of Brownian motion in Rd and renormalization

Double intersection local times [alpha](x,.) of Brownian motion which measure the size of the set of time pairs (s, t), s [not equal to] t, for which Wt and Ws + x coincide can be developed into series of multiple Wiener-Ito integrals. These series representations reveal on the one hand the degree of smoothness of [alpha](x,.) in terms of eventually negative order Sobolev spaces with respect to the canonical Dirichlet structure on Wiener space. On the other hand, they offer an easy access to renormalization of [alpha](x,.) as x --> 0. The results, valid for any dimension d, describe a pattern in which the well known cases d = 2, 3 are naturally embedded.

Interpolation, correlation identities, and inequalities for infinitely divisible variables

We present an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables. This is then applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities.

On free and classical type G distributions

There is a one-to-one correspondence between classical one-dimensional infinitely divisible distributions and free infinitely divisible distributions. In this work we study the free infinitely divisible distributions corresponding to the one-dimensional type  distributions. A new characterization of classical type  distributions is given first and the class of type  classical infinitely divisible distributions is introduced. The corresponding free type  distributions are studied and the role of a special symmetric beta distribution is shown as a building block for free type  distributions. It is proved that this symmetric beta distribution is the free multiplicative convolution of an arcsine distribution with the Marchenko-Pastur distribution.

Empirical probability generating function: An overview

Abstract A convenient approach to the statistical analysis of distributions for counts is possible using the empirical probability generating function. In this paper we give an overview of recent results and show the usefulness and advantages of this methodology. On one hand, there are some stochastic models in which the probability generating function arises naturally and therefore it seems reasonable to use its empirical counterpart. On the other hand, this statistical tool has demonstrated to be useful in the study of classical statistical problems of distributions for counts, especially in exploratory data analysis, rapid multi-parameter estimation and testing the goodness of fit. Our recommendation is to make allowance for the empirical probability generating function when dealing with statistical inference for discrete distributions.

Stationary and self-similar processes driven by Lévy processes

Using bivariate Levy processes, stationary and self-similar processes, with prescribed one-dimensional marginal laws of type G, are constructed. The self-similar processes are obtained from the stationary by the Lamperti transformation. In the case of square integrability the arbitrary spectral distribution of the stationary process can be chosen so that the corresponding self-similar process has second-order stationary increments. The spectral distribution in question, which yields fractional Brownian motion when the driving Levy process is the bivariate Brownian motion, is shown to possess a density, and an explicit expression for the density is derived.

A class of multivariate infinitely divisible distributions related to arcsine density

Two transformations A1 and A2 of Levy measures on R d based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of A1 and A2 are determined and it is shown that they have the same range. The class of infinitely divisible distributions on R d with Levy measures being in the common range is called the class A and any distribution in the class A is expressed as the law of a stochastic integral � 1 0 cos(2 −1 πt)dXt with respect to a Levy process {Xt }. This new class includes as a proper subclass the Jurek class of distributions. It is shown that generalized type G distributions are the image of distributions in the class A under a mapping defined by an appropriate stochastic integral. A2 is identified as an Upsilon transformation, while A1 is shown not to be.

Large deviations for a class of chaos expansions

The purpose of this paper is to present a general extended contraction principle for large deviations and apply it to obtain large deviations for random variables having chaos developments of exponential type.

INDEX OF EFFECTIVENESS OF A MULTIVARIATE ENVIRONMENTAL MONITORING NETWORK

In an attempt to design an optimal network for monitoring several hydrocarbons in Mexico City, this paper introduces an index of effectiveness of a multi-monitoring environmental network. The index quantifies the possibility of choosing a network design from a given set of available sites to monitor in an optimal way several variables. In addition the paper reviews the statistical techniques used in re-designing and evaluating an operating environmental monitoring network.

Exploratory data analysis for counts using the empirical probability generating function 1

We present a graphical method based on the empirical probability generating function for preliminary statistical analysis of distributions for counts. The method is especially useful in fitting a Poisson model, or for identifying alternative models as well as possible outlying observations from general discrete distributions.

COVARIANCE-PARAMETER LÉVY PROCESSES IN THE SPACE OF TRACE-CLASS OPERATORS

The paper deals with Levy processes with values in L1(H), the Banach space of trace-class operators in a Hilbert space H. Levy processes with values and parameter in a cone K of L1(H) are introduced and several properties are established. A family of L1(H)-valued Levy processes is obtained via the subordination of K-parameter, L1(H)-valued Levy processes, identifying explicitly their generating triplets.