Product Disintegrations: Some Examples
We provide a collection of examples involving the concept of a product disintegration, which generalizes exchangeability.
math.PR↗
arXiv subjects
Rafael Rigão Souza
Publications and source records attributed to Rafael Rigão Souza.
Thermodynamic formalism for general iterated function systems with measures
This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSm given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional.
math.DS↗
Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure
Consider a compact metric space $(M, d_M)$ and $X = M^{\mathbb{N}}$. We prove a Ruelle's Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in [Bull. Braz. Math. Soc. 45 (2014), pp. 53-72] which are defined from a continuous function $A : M \times M \to \mathbb{R}$ that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle $S^1$. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.
math.DS↗
A characterization of the strong law of large numbers for Bernoulli sequences
The law of large numbers is one of the most fundamental results in Probability Theory. In the case of independent sequences, there are some known characterizations; for instance, in the independent and identically distributed setting it is known that the law of large numbers is equivalent to integrability. In the case of dependent sequences, there are no known general characterizations -- to the best of our knowledge. We provide such a characterization for identically distributed Bernoulli sequences in terms of a product disintegration.
math.PR↗
A Dynamic Model for Double Bounded Time Series With Chaotic Driven Conditional Averages
In this work we introduce a class of dynamic models for time series taking values on the unit interval. The proposed model follows a generalized linear model approach where the random component, conditioned on the past information, follows a beta distribution, while the conditional mean specification may include covariates and also an extra additive term given by the iteration of a map that can present chaotic behavior. The resulting model is very flexible and its systematic component can accommodate short and long range dependence, periodic behavior, laminar phases, etc. We derive easily verifiable conditions for the stationarity of the proposed model, as well as conditions for the law of large numbers and a Birkhoff-type theorem to hold. A Monte Carlo simulation study is performed to assess the finite sample behavior of the partial maximum likelihood approach for parameter estimation in the proposed model. Finally, an application to the proportion of stored hydroelectrical energy in Southern Brazil is presented.
math.ST↗
An Ergodic Theorem on Ergodic Transport
Here we present an ergodic theorem which adapts a Theorem by J. Elton to the classical thermodynamical formalism and to ergodic transport. First, we discuss how Elton's theorem can be used to characterise Gibbs measures for expanding maps. Such characterisation will be done by constructing a stochastic process, defined by a iterated function system (IFS), whose empirical measure converges to the Gibbs measure, in the sense that the mean of any test function evaluated in the outcomes of this stochastic process converges to the integral of such test function with respect to the Gibbs measure. In this way we present a stochastic algorithm that compute integrals of functions. After this, we turn our attention to ergodic transport: given two sets $X$ and $Ω$, a measure $μ$ on $X$ and a dynamics $T$ on $Ω$, we consider the set of probability measures on $X \times Ω$ whose projections on the second coordinate are $T$-invariant, while the projections on the first coordinate are $μ$. Such measures are called transport plans. We call Gibbs plan any transport plan that maximizes a pressure functional that is defined by a potential function added to an entropy term. As in the classical thermodynamical formalism case, we adapt Elton's theorem to define a stochastic process (using a IFS) whose empirical measures converges to the Gibbs plan. We provide examples and show explicitly calculations in the case where $X$ has two elements and the cost function depends on the two first coordinates of $Ω$.
math.DS↗