L’objectif de cet exposé, basé sur un travail commun avec Matija Vidmar, est d’étudier la classe des processus de Markov positifs qui admettent une relation de dualité de Laplace : les transformées de Laplace des processus sont reliées en échangeant les rôles de l’argument et de l’état initial. Ce type de dualité apparaît naturellement, par exemple, dans des systèmes présentant des phénomènes de branchement. Au-delà du cadre classique du branchement, nous montrons qu’une grande variété de processus et de générateurs sont en dualité de Laplace.

Dans un premier temps, d’un point de vue théorique, nous établissons qu’un processus admet un dual de Laplace si et seulement si son semi-groupe laisse invariant l’espace des fonctions complètement monotones (sous réserve de conventions pour 0 × ∞ et ∞ × 0). D'un point de vue plus constructif, nous identifions ensuite sept briques fondamentales à partir desquelles une telle dualité peut être construite. Les processus associés peuvent être vus comme des généralisations des processus de branchement à état continu et incluent plusieurs modèles — parfois introduits indépendamment de la dualité — utilisés pour représenter des environnements aléatoires, l’immigration, la compétition et d’autres dynamiques. Un outil analytique central, développé ici dans un cadre général et unificateur, est la notion de symbole de Laplace associé à un générateur.

Robust estimation of the covariance matrix and detection of outliers remain major challenges in statistical data analysis, particularly when the proportion of contaminated observations increases with the size of the dataset. Outliers can severely bias parameter estimates and induce a masking effect, whereby some outliers conceal the presence of other outliers, further complicating their detection. Although many approaches have been proposed for covariance estimation and outlier detection, to our knowledge, none of these methods have been implemented in an online setting. In this paper, we focus on online covariance matrix estimation and outlier detection. Specifically, we propose a new method for simultaneously and online estimating the geometric median and variance, which allows us to calculate the Mahalanobis distance for each incoming data point before deciding whether it should be considered an outlier. To mitigate the masking effect, robust estimation techniques for the mean and variance are required. Our approach uses the geometric median for robust estimation of the location and the median covariance matrix for robust estimation of the dispersion parameters. The new online methods proposed for parameter estimation and outlier detection allow real-time identification of outliers as data are observed sequentially. The performance of our methods is demonstrated on simulated datasets.

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We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a “stationarized” representation, making it particularly well-suited for time-series analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula à la Fawcett for its expected value in terms of Magnus expansions. This is joint work with Eduardo Abi Jaber.

Survival analysis is often applied to sensitive data making privacy protection of crucial importance. In this talk, we consider local differential privacy in survival analysis using private failure indicators. First, I will provide an overview of existing methods and discuss how individual privacy can be preserved. Next, I will introduce a non-parametric kernel estimator for the cumulative hazard function using a locally differentially private mechanism on failure indicators. Under mild conditions, we also prove lower bounds on the minimax rates of convergence and show that our estimator is minimax optimal under well-chosen bandwidths. Finally, I will illustrate our estimator with numerical results.

The critical Fortuin–Kasteleyn random planar map with parameter q>0 is a model of random (discretised) surfaces decorated by loops, related to the q-state Potts model. For q<4, Sheffield established a scaling limit result for these discretised surfaces, where the limit is described by a so-called Liouville quantum gravity surface decorated by a conformal loop ensemble. At q=4 a phase transition occurs, and the correct rescaling needed to obtain a limit has so far remained unclear. I will talk about joint work with William Da Silva, XinJiang Hu, and Mo Dick Wong, where we identify the right rescaling at this critical value and prove a number of convergence results.