In this talk a review of the subject of non-singulariy of Ginibre and Wigner random matrices will be presented. Then we will present recent results about the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, Generalized, universal and banded random matrices. We will present universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. We will show the important role played by a concentration function inequality due to Kesten that allows us to improve known universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.
This is joint work with Paulo Manrique and Rahul Roy.

In this paper we present a novel inference methodology to perform Bayesian inference for Cox processes in space and/or time where the intensity function depends on a multivariate Gaussian process. The novelty of the method lies on the fact that no discretisation error is involved despite the non- tractability of the likelihood function and infinite dimensionality of the problem. The method is based on a Markov chain Monte Carlo algorithm that samples from the joint posterior distribution of the parameters and latent variables of the model. A particular choice of the dominating measure to obtain the likelihood function corrects previous attempts to solve the problem in an exact framework. The models allow the use of covariates to explain the dynamics of the intensity function. Simulated examples illustrate the methodology and compare different alternatives for some of the MCMC steps. This is joint work with Dani Gamerman.

In dynamical systems one usually considers the dynamics of “typical diffeomorphisms”. Of course, one of the very first questions is to define “typical”! Pioneers used Baire category: countable intersections of open and dense sets. Later, Kolmogorov suggested to use the concept which is called today “prevalence”: some kind of substitute for the Lebesgue measure in infinite dimension. In this talk, I will begin by explaining the advantages and drawbacks of these two notions. Then, I will restrict myself to the 1 dimensional case and discuss the Malliavin-Shavgulidze measure on the group of diffeomorphisms of the circle, related to the Brownian motion. It will be a pleasant opportunity to advertise part of the PhD thesis of my latest student: Michele Triestino. One would like to understand the dynamics of almost all diffeomorphism of the circle, with respect to this Malliavin-Shavgulidze probability.

As stated in any textbook, Thermodynamics is the field of Science devoted to the study of relations between macroscopic observables of a system such as heat, work, energy. The microscopic understanding of the macroscopic laws that Thermodynamics provide us with was finally achieved by means of the application of probabilistic concepts to mechanical systems within the Statistical Mechanics approach and the assumption of the macroscopic (Thermodynamic) limit. However, as technology has moved on, interesting systems have downsized and one has started facing the study of heat, energy and work relations clearly off the thermodynamical limit. Although the (standard) macroscopic laws of Thermodynamics are thus crippled, it is possible to establish equivalent relations which allow predicting the behaviour of physical quantities such as the injected (dissipated) power into (out of) the system, the heat flux within it as well as several other fluctuation relations.

Along these lines, I will present some results on the thermostatistical properties of small in- and out-of-equilibrium massive systems subject to non-linear potentials and in contact with Gaussian and non-Gaussian reservoirs with the context of the Lévy-Itô theorem. A typical example of thermostats of the latter ilk is the Poissonian (shot-noise) heat bath that can be regarded as a means of describing the energy input to particles by ATP hydrolysis – a phenomenon that can be found in molecular motors. A special emphasis to the physical significance of higher than two statistical cumulants of non-Gaussian reservoirs will be given. Moreover, it will be shown that they can be interpreted as supplementary heat sources.

We discuss joint work with M. E. Vares concerning a “random walk” on Z whose jump rates depend on an underlying contact process in (supercritical) upper equilibrium. We show an invariance principle, though without finding an i.i.d. regenerative structure.

A typical stochastic process has infinite memory in the sense that its conditional distribution at time 0 depends on the whole infinite past. In this talk we consider a class of processes, in discrete time and space, where this distribution can be approximated arbitrarily well by looking at finite portions of the past. These processes are represented by “mixtures of context trees,” and coincide with processes with almost surely continuous transition probabilities. As such, they generalize well-known classes of processes in the literature, such as finite-order Markov chains, context tree processes and regular g measures.

Gaussian graphical models have been extensively used to model conditional independence via the concentration matrix of a random vector. They are particularly relevant to incorporate structure when the length of the vector is large and naive methods lead to unstable estimation of the concentration matrix. In covariance selection, we have a latent network among vector components such that two components are not connected if they are conditionally independent, that is, if their corresponding entry in the concentration matrix is zero. In this work, we expect that, in addition, vector components show a block dependency structure that represents community behavior in the context of biological and social applications, that is, connections between nodes from different blocks are sparse while connections within nodes of the same block are dense. Thus, to identify the latent network and detect communities, we propose a Bayesian approach with a hierarchical prior in two levels: a spike-and-slab prior on each off-diagonal entry of the concentration matrix for variable selection; and a degree-corrected stochastic blockmodel to capture community behavior. To conduct inference, we develop an efficient routine based on ridge regularization and MAP estimation. Finally, we demonstrate the proposed approach in a meta-genomic dataset of complex microbial biofilms from dental plaque and show how bacterial communities can be identified. This is joint work with Lijun Peng.

Consideramos um modelo de passeio aleatório em um meio aleatório dinâmico dado por um sistema Poissoniano de passeios aleatórios independentes em equilíbrio, para o qual mostramos uma lei dos grandes números e um teorema central do limite.