## Seminários de Probabilidade 2023 – 2º Semestre

• ##### English
###### 11/12 (presencial)

We study Bayesian approaches to causal inference via propensity score regression. Much of Bayesian methodology relies on parametric and distributional assumptions, with presumed correct specification, whereas the extant propensity score methods in Bayesian literature have relied on approaches that cannot be viewed as fully Bayesian in the context of conventional ‘likelihood times prior’ posterior inference. We emphasize that causal inference is typically carried out in settings of mis-specification, and develop strategies for fully Bayesian inference that reflect this. We focus on methods based on decision-theoretic arguments, and show how inference based on loss-minimization can give valid and fully Bayesian inference. We propose a computational approach to inference based on the Bayesian bootstrap which has good Bayesian and frequentist properties.

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###### 27/11 (remoto)

This talk will present an overview of the behavior of the eigenvalues of the fractional Brownian matrix motion and other related matrix processes. We emphasize on a possible extension of the Dyson-Brownian motion, namely the dynamics of the eigenvalues processes and their non-colliding property, the limit of the associated empirical process, as well as the free Brownian motion and the non commutative fractional Brownian motion.

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###### 06/11 (remoto)

If Y is a random vector in R^d we denote by P_Y its probability distribution. Consider a random variable X and a d-dimensional random vector Y. We develop a multidimensional variant of the Stein-Malliavin calculus which allows to measure the Wasserstein distance between the law P_(X, Y) and the probability distribution P_Z x P_Y, where Z is a Gaussian random variable. That is, we give estimates, in terms of the Malliavin operators, for the distance between the law of the random vector (X, Y) and the law of the vector (Z,Y), where Z is Gaussian and independent of Y. Then we focus on the particular case of random vectors in Wiener chaos and we give an asymptotic version of this result. In this situation, this variant of the Stein-Malliavin calculus has strong and unexpected consequences.

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###### 30/10 (remoto)

In this talk we study the scaling limit of a random field which is a non-linear transformation of the gradient Gaussian free field. More precisely, our object of interest is the recentered square of the norm of the gradient Gaussian free field at every point of the square lattice. Surprisingly, in dimension 2 this field bears a very close connection to the height-one field of the Abelian sandpile model studied in Dürre (2009). In fact, with different methods we are able to obtain the same scaling limits of the height-one field: on the one hand, we show that the limiting cumulants are identical (up to a sign change) with the same conformally covariant property, and on the other that the same central limit theorem holds when we view the interface as a random distribution. We generalize these results to higher dimensions as well.

Joint work with Rajat Subhra Hazra (Leiden), Alan Rapoport (Utrecht) and Wioletta Ruszel (Utrecht).

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###### 16/10 (presencial)

Increasing trees are rooted trees, where each vertex has a unique label and the labels along paths away from the root are in increasing order. A Random recursive tree on n vertices (abbreviated as RRTs) is a tree chosen uniformly at random from the set of increasing trees with vertex labels {1,…,n}. The idea of cutting random recursive trees was introduced by Meir and Moon in 1974. They studied the following procedure: Start with a random recursive tree on n vertices. Choose an edge at random and remove it, along with the cut subtree that does not contain the root. Repeat until the remaining tree consists only of the root; at which point, we say that the tree has been deleted.

Let X be the number of edge removals needed to delete a RRT with n vertices. The random variable X has been thoroughly studied and analogous variables under distinct models of random trees have been analyzed; in particular, X grows asymptotically as n ln(n). In this talk we propose and study a method for cutting down a random recursive tree that focuses on its largest degree vertices. Enumerate the vertices of a random recursive tree of size n according to a decreasing order of their degrees. The targeted, vertex-cutting process is performed by sequentially removing vertices according to that order and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed.

Joint work with Laura Eslava and Marco L. Ortiz.

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###### 02/10 (presencial)

In this joint work with Leonardo Rolla we study a one-dimensional contact process with two infection parameters. One of these parameters gives the infection rates at the boundaries of a finite infected region and the other one gives the rates within that region. We prove that the critical value of each of these parameters is a strictly monotone continuous function of the other parameter. We also show that if one of these parameters is equal to the critical value of the standard contact process and the other parameter is strictly larger, then the infection starting from a single point has a positive probability of surviving.

###### 25/09 (presencial)

A stochastic adding machine (defined in [PR Killeen and T. J. S. Taylor, Nonlinearity 13 (2000), no. 6, 1889–1903]) is a Markov chain whose states are natural integers, which models the process of adding the number \$1\$ but where there is a probability of failure in which a carry is not performed when necessary.
In this lecture, we will talk about probabilistic properties of extensions for the stochastic adding machine and their connections with other areas of mathematics such as Complex Dynamics and Linear Dynamics.
This is a joint work with Danilo Caprio and Glauco Valle.

###### 18/09 (presencial)

Pablo Ferrari and Luiz Renato Fontes introduced The Random Average Process (RAP) in 98. We are interested in the discrete-time version of the RAP. This process is a random surface whose heights evolve taking a convex combination of the previous heights. In this dynamic, a random matrix of probabilities with independent and identically distributed rows determines the weights of the convex combinations. The process seen from the height in the origin is the random surface result of subtracting the height in the origin to all the heights in the initial surface. Under certain conditions, Ferrari and Fontes proved in 98 the existence of a limit process for the process seen from the height of the origin. In this talk I will discuss a Central Limit Theorem in the spacial variable for the limit process seen from the height in the origin. This is a joint work with Luiz Renato Fontes and Leonel Zuaznabar.

###### 04/09 (presencial)

Recall that in the simple Random Walk (RW) on Z the walker, starting at 0, just jumps either to the right or to the left with the same probability. It is a classical result that the simple RW on Z is recurrent. In the Edge-Reinforced Random Walk (ERRW) the walker keeps track of the edges already visited and gives extra bias to the edges mostly visited. We would expect that the behavior of the ERRW depends on the strength of the extra bias we decide to give to the edges. The ERRW is a non-markovian process introduced by Diaconis and the first results about it goes back to Davis in 1990. Davis showed, under some assumptions, that the ERRW on Z is either recurrent or it localizes in a single edge with probability 1. What would happen if instead of a single ERRW we consider two or more walkers reinforcing the edges of Z? In an ongoing project, together with Nina Gantert (TUM) and Fabian Michel (TUM), we plan to answer the above question.

###### 21/08 (presencial)

It is known for expanding dynamical systems and finite state Markov chains that the asymptotic behaviour of the minimal distance between two orbits up to time n is given by its correlation dimension.

In this talk, we will discuss this problem in a randomized setting with not necessarily expanding fibres. If the fibres and the basis of the random system under consideration are sufficiently mixing, then a similar but more complex result holds: there are two relevant dimensions and, depending on the stochastic process in the basis, either one or the other is dominant. In particular, there is a phase transition, which is unknown in the framework of a classical dynamical system.

Joint work with Jerome Rousseau and Sebastien Gouezel. For the preprint, see https://hal.science/hal-03788538v1

###### 14/08 (presencial)

The renewal contact process, introduced in 2019 by Fontes, Marchetti, Mountford, and Vares, extends the Harris contact process by allowing the possible cure times to be determined according to independent renewal processes (with some interarrival distribution mu) and keeping the transmission times determined according to independent exponential times with a fixed rate lambda. In this talk we will discuss conditions on mu to have a positive and finite critical parameter in the renewal contact process. Joint work with Maria Eulalia Vares.

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