Seminários de probabilidade – Segundo Semestre de 2025
Quando forem online, as palestras ocorrerão via Google Meet às segundas-feiras às 15h30, a menos de algumas exceções devidamente indicadas.
Quando forem presenciais, as palestras ocorrerão na sala C-116 às segundas-feiras às 15h30, a menos de algumas exceções devidamente indicadas.
Todas as palestras são em inglês.
Lista completa (palestras futuras podem sofrer alterações)
We discuss convergence results of renormalised processes and give a simplified approach to convergence of various models introduced by Toth.
De Finetti’s Theorem states that all N-indexed exchangeable sequences of real-valued random variables are mixings of i.i.d sequences. For real-valued random processes with exchangeable increments on [0, 1], Kallenberg’s 1973 result provides a complete characterisation of these processes via another mixing relationship. Continuum random trees are random tree-like metric spaces that arise naturally as scaling limits of various models of discrete random trees. In this talk, we will focus in particular on two subclasses of continuum random trees: the so-called stable trees and inhomogeneous continuum random trees. An analogue of Kallenberg’s Theorem for continuum random trees first appeared as a claim in a 2004 paper by Aldous, Miermont and Pitman. They suggested that, in much the same way that a stable bridge process on [0, 1] is a mixing of certain extremal exchangeable processes, stable trees are mixings of inhomogeneous continuum random trees.We present an outline of a rigorous argument supporting this claim, based on a novel construction that applies to both classes of trees. We will also briefly discuss some implications of this result on critical random graphs.
The talk is based on the paper Stables trees as mixings of inhomogeneous continuum random trees, Stochastic Processes and their Applications 175, 2024, 104404.
Link google meet: https://meet.google.com/idy-jhss-htx
In this talk, we investigate the phenomenon of coming-down from infinity for subcritical cooperative branching processes with pairwise interactions under suitable conditions. A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. In particular, we are interested in the speed of BPI-processes when their initial population is very large, as well as in their second order fluctuations. This is a joint work with Gabriel Berzunza (University of Liverpool).
The Josephus problem is a well–studied elimination problem consisting in determining the position of the survivor after repeated applications of a deterministic rule removing one person at a time from a given group. In this talk, we introduce a natural probabilistic variant of this process. More precisely, in this variant, the survivor is determined after performing a succession of Bernouilli trials with parameter p designating each time the person to remove. When the number of participants tends to infinity, the main result characterises the limit distribution of the position of the survivor with an increasing degree of precision as the parameter approaches the unbiased case p=1/2. Then, the convergence rate to the position of the survivor is obtained in the form of a Central-Limit Theorem.A number of other variants of the suggested probabilistic elimination process are also considered. They each admit a specific limit behavior which, in most cases, is stated in the form of an open problem.
In this work, in collaboration with Blumenthal and Taylor-Crush, I present a generalization of a fundamental result, the Gerschgorin circle theorem, to obtain enclosures of the discrete spectrum of a transfer operator preserving a strong Banach space compactly embedded in a weak Banach space. The enclosures are obtained by rigorously bounding the weak resolvent norm of a finite rank approximation of the transfer operator. This result has important consequences, allowing us to understand the finer statistical properties of systems satisfying a Lasota-Yorke inequality, as uniformly expanding maps and systems with additive noise by enclosing the spectrum of the associated Markov operators.
Given a Baire metric space $(Y, d)$, a continuous map $T : Y \to Y$ and $\varphi$ a continuous bounded real valued function of $Y$, the set of $(T,\varphi)$-irregular points, also points with historic behavior, is formed by those points whose Cesàro average does not converge. The Birkhoff's ergodic theorem ensures that, for any Borel $T$-invariant probability measure $\mu$ and every $\mu$-integrable observable $\varphi: Y \to \mathbb R$, the sequence of averages converges at $\mu$-almost every point in $Y$. So, the set of irregular points is negligible with respect to any $T$-invariant probability measure. In the last decades, though, there has been an intense study concerning the set of points for which Cesàro averages do not converge. Contrary to the previous measure-theoretical description, the set of the irregular points may be Baire generic. We introduce a notion of sensitivity with respect to a continuous real-valued bounded map which provides a sufficient condition for a continuous transformation, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior. This is a joint work with M. Carvalho (CMUP), V. Coelho (UFOB) and P. Varandas (UFBA/Univ. de Aveiro).
References: M. Carvalho, V. Coelho, L. Salgado and P. Varandas, Sensitivity and historic behavior for continuous maps on Baire metric spaces. 2024;44(1):1-30.