Séminaire Probabilités et Statistique

Hawkes processes are a family of point processes for which the occurrence of any event increases the probability of further events occurring. Although the linear Hawkes process, for which a representation in the form of a superposition of branching processes exists, is particularly well studied, difficulties remain in estimating the parameters of the process from imperfect data (noisy, missing or aggregated data), since the usual estimation methods based on maximum likelihood or least squares do not necessarily offer theoretical guarantees or are numerically too costly.
In this work, we propose a spectral approach well-adapted to this context, for which we prove consistency and asymptotic normality. In order to derive these properties, we show that Hawkes processes can be studied through the scope of mixing, opening the use of central limit theorems that already exist in the literature.
I will then present two applications of this approach: to aggregated data (joint work with Gabriel Lang); and to noisy data (joint work with Anna Bonnet, Miguel Martinez and Maxime Sangnier).

Après avoir introduit les U-statistiques de données indépendantes, le théorème limite central fonctionnel et la loi des grands nombres correspondantes, nous présenterons des résultats récents pour des U-statistiques basées sur des suites stationnaires dépendantes à valeurs dans un espace métrique séparable. Nous traiterons le cas des suites beta-mélangeantes (qui se prêtent bien au couplage) ainsi que celui des U-statistiques à valeurs dans un espace de Hilbert. Ces dernières se révèlent utiles dans certains tests statistiques.

The stochastic dynamic matching problem has recently drawn attention in the stochastic-modeling community due to its numerous applications, ranging from supply-chain management to kidney exchange programs. In this presentation, we consider a matching problem in which items of different classes arrive according to independent Poisson processes, unmatched items are stored in a queue, and compatibility constraints are described by a simple graph on the classes, so that two items can be matched if their classes are neighbors in the graph. We analyze the efficiency of matching policies, not only in terms of system stability, but also in terms of matching rates between different classes. Our results rely on the observation that, under any stable policy, the matching rates satisfy a conservation equation that equates the arrival and departure rates of each item class.

This presentation is based on a joint work with Fabien Mathieu (LINCS) and Ana Bušić (Inria and PSL University). A preprint is available at the following address: https://arxiv.org/abs/2112.14457.

Imagine a collection of randomly distributed points in space, and build a graph on it according to some rules. We now take the sum of all edge lengths and want to know if this sum verifies a central limit theorem. In our context, the random collection of points is given by a Poisson measure and the sum of edge lengths is an example of a Poisson functional. In this talk, using the so-called Malliavin-Stein method, we show a result which allows us to derive central limit theorems, as well as speeds of convergence, for functionals of general Poisson measures, under minimal moment assumptions. Our main application is the closure of a conjecture about a convergence result of the Online Nearest Neighbour Graph.

Le Processus de saut renforcé par sommet (VRJP en raison de son acronyme anglais) est un processus aléatoire en temps continu auto-renforcé défini sur un graphe : plus il passe par un sommet, plus il est probable qu’il y revienne à l’avenir. Depuis les travaux de Sabot et Tarrès, on sait que le VRJP peut être vu comme un processus Markovien en environnement aléatoire. Cet environnement est caractérisé par un certain champ aléatoire U qui, étonnamment, était déjà connu par les physiciens en tant que modèle sigma super-symétrique H^{(2|2)}, un objet important en physique quantique. Par un changement de variable, on peut ramener l’étude du champ U à celle d’un autre champ beta. Ce champ beta permet de définir un opérateur de Schrödinger aléatoire H dont les propriétés sont reliées au VRJP. H est notamment inversible et son inverse G permet de retrouver l’environnement du VRJP. Dans cet exposé, nous expliquerons ce cheminement allant du VRJP vers les champs aléatoires et nous expliquerons les résultats connus sur ces champs et sur le VRJP. Pour finir, nous parlerons d’un résultat plus récent établissant la limite d’échelle de G sur le cercle. Cela permet de définir un opérateur de Schrödinger aléatoire continu analogue à H sur le cercle.

In the study of singular SPDEs, it has been a challenging problem to obtain a simple proof of a general probabilistic convergence result (BPHZ theorem). Differently from Chandra and Hairer’s Feynman diagram approach, Linares, Otto, Tempelmayr, and Tsatsoulis recently proposed an inductive proof based on the spectral gap inequality by using their multiindex language. Inspired by their approach, Hairer and Steele also obtained an inductive proof by using the regularity structure language. In this talk, we introduce an extension of the regularity structure including integrability exponents and provide a simpler proof of BPHZ theorem.
This talk is based on a joint work with Ismael Bailleul (Univ Brest).

Dormancy is a complex trait that has evolved independently many times across the tree of life. In particular many micro-organisms can enter a reversible state of vanishing metabolic activity. The corresponding dormancy periods can range from a few hours to potentially thousands of years. Also, the dormancy transitioning mechanisms are highly diverse, including spontaneous dormancy initiation and resuscitation, responsive switching due to environmental cues, and competition-induced dormancy initiation. In general, dormancy allows a population to maintain a reservoir of genotypic and phenotypic diversity (that is, a seed bank) that can contribute to its longterm survival and coexistence. In this talk, we review some of the probabilistic structures that emerge from stochastic individual based models involving dormancy.

Recent works in time-frequency analysis proposed to switch the focus from the maxima of the spectrogram toward its zeros, which form a random point pattern with a very stable structure. Several signal processing tasks, such as component disentanglement and signal detection procedures, have already been renewed by using modern spatial statistics onthe pattern of zeros. Tough, they require cautious choice of both the discretization strategy and the observation window in the time-frequency plane. To overcome these limitations, we propose a generalized time-frequency representation: the Kravchuk transform, especially designed for discrete signals analysis, whose phase space is the unit sphere, particularly amenable to spatial statistics. We show that it has all desired properties for signal processing, among which covariance, invertibility and symmetry, and that the point process of the zeros of the Kravchuk transform of complex white Gaussian noise coincides with the zeros of the spherical Gaussian Analytic Function. Elaborating on this theorem, we finally develop a Monte Carlo envelope test procedure for signal detection based on the spatial statistics of the zeros of the Kravchuk spectrogram.

After reviewing the unorthodox path focusing on the zeros of the standard spectrogram and the associated theoretical results on the distribution of zeros in the case of white noise, I will introduce the Kravchuk transform and study the random point process of its zeros from a spatial statistics perspective. Then I will present the designed Monte Carlo envelop test, and illustrate its numerical performance in adversarial settings, with both low signal-to-noise ratio and small number of samples, and compare it to state-of-the-art zeros-based detection procedures.

Dans cet exposé, on s’intéressera à une marche aléatoire non markovienne, telle que la probabilité que la marche aille en un point donné est plus faible si elle est déjà passée souvent le long de l’arête entre la position initiale et la cible. Les modèles de ce type les plus étudiés sont ceux dans lesquels les arêtes sont non orientées. Cependant, en 2008 Tóth et Vető ont introduit une telle marche aléatoire avec auto-interactions pour laquelle les arêtes sont orientées, et découvert des propriétés très différentes de celles des modèles avec arêtes non orientées. Malgré l’intérêt d’un tel comportement, très peu de résultats ont été obtenus depuis sur ce modèle. On présentera de nouveaux théorèmes limites pour cette marche aléatoire.