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Jean-François Delmas

Publications and source records attributed to Jean-François Delmas.

At least 19 recordsLinked to original sources
arXiv2026

Coupling some conditioned L{é}vy trees with the Kesten tree

We consider locally compact L{é}vy trees conditioned to be large, with respect to different criterion: its height, its maximal ''size'' vertex and its total ''mass''. In the critical case, we provide a coupling with a truncated Kesten tree which then allows to directly prove the local convergence in distribution of the conditioned L{é}vy tree to be large towards the Kesten tree. We also consider the sub-critical and super-critical cases. In the former case the results can be partial, due to a possible condensation phenomenon which is outside the mathematical framework used in this paper.

math.PR↗
arXiv2026

Site Frequency Spectrum in stationary branching populations

This paper explores the Site Frequency Spectrum (SFS) in stationary branching populations. We derive estimates for the SFS associated with a sample from a continuous-state branching process conditioned to never go extinct, utilizing a quadratic branching mechanism. The genealogy of such processes is represented by a real tree with a semi-infinite branch, and we compute the expectation of the SFS under the infinitely-many-sites assumption as the sample size approaches infinity. Additionally, we present a continuum version of the SFS as a random point measure on the positive real line and compute the density of its expected measure explicitly. Finally, we derive estimates for the size of the clonal subpopulation carrying the same genotype as the most recent common ancestor of the whole population at a given time.

math.PR↗
arXiv2026

Csiszár indices and interpolating copulas

We study various properties of $f$-divergences and Csiszár indices between two probability distributions in very general setups for the convex function $f$ and for the probability distributions. We establish general structural properties of $f$-divergences and show how they are inherited by the associated Csiszár indices, including monotonicity and invariance under suitable transformations. We also study the relationship between Csiszár indices and copula representations of random vectors. When the marginal distributions have atoms, the copula representation is not unique and the Csiszár index of the transformed vectors may increase. We build a large family of interpolating copulas which minimize the Csiszár index and thus preserve the dependence structure of the initial vector.

math.ST↗
arXiv2025

Equivalence by coupling for heterogeneous SIS models*

We consider the optimal allocation of (perfect) vaccine in an heterogeneous SIS model. Using a coupling approach, we explain how different models for the heterogeneity of the population lead to the same Pareto frontier in the cost/loss valuation of the vaccinations strategies. This covers in particular the elementary continuous representation of discrete models and the measure preserving transformation which appears in graphon theory.

math.PR↗
arXiv2025

Off-the-grid learning of mixtures from a continuous dictionary

We consider a general non-linear model where the signal is a finite mixture of an unknown, possibly increasing, number of features issued from a continuous dictionary parameterized by a real non-linear parameter. The signal is observed with Gaussian (possibly correlated) noise in either a continuous or a discrete setup. We propose an off-the-grid optimization method, that is, a method which does not use any discretization scheme on the parameter space, to estimate both the non-linear parameters of the features and the linear parameters of the mixture. We use recent results on the geometry of off-the-grid methods to give minimal separation on the true underlying non-linear parameters such that interpolating certificate functions can be constructed. Using also tail bounds for suprema of Gaussian processes we bound the prediction error with high probability. Assuming that the certificate functions can be constructed, our prediction error bound is up to $\log$-factors similar to the rates attained by the Lasso predictor in the linear regression model. We also establish convergence rates that quantify with high probability the quality of estimation for both the linear and the non-linear parameters. We develop in full details our main results for two applications: the Gaussian spike deconvolution and the scaled exponential model.

stat.ML↗
arXiv2025

Topological Properties of the Effective Reproduction Number in an Heterogeneous SIS Model

This present results lay the foundations for the study of the optimal allocation of vaccine in the simple epidemiological SIS model where one consider a very general heterogeneous population. In the present setting each individual has a type x belonging to a general space, and a vaccination strategy is a function $η$ where $η$(x) $\in$ [0, 1] represents the proportion of non-vaccinated among individuals of type x. We shall consider two loss functions associated to a vaccination strategy $η$: either the effective reproduction number, a classical quantity appearing in many models in epidemiology, and which is given here by the spectral radius of a compact operator that depends on $η$; or the overall proportion of infected individuals after vaccination in the maximal endemic state.By considering the weak-* topology on the set $Δ$ of vaccination strategies, so that it is a compact set, we can prove that those two loss functions are continuous using the notion of collective compactness for a family of operators. We also prove their stability with respect to the parameters of the SIS model. Eventually, we consider their monotonicity and related properties in particular when the model is ``almost'' irreducible.

q-bio.PE↗
arXiv2025

Optimal Vaccination Strategies for an Heterogeneous SIS Model

We study in a general mathematical framework the optimal allocation of vaccine in an heterogeneous population. We cast the problem of optimal vaccination as a bi-objective minimization problem min(C($η$),L($η$)), where C and L stand respectively for the cost and the loss incurred when following the vaccination strategy $η$, where the function $η$(x) represents the proportion of non-vaccinated among individuals of feature x.To measure the loss, we consider either the effective reproduction number, a classical quantity appearing in many models in epidemiology, or the overall proportion of infected individuals after vaccination in the maximal equilibrium, also called the endemic state. We only make few assumptions on the cost C($η$), which cover in particular the uniform cost, that is, the total number of vaccinated people.The analysis of the bi-objective problem is carried in a general framework, and we check that it is well posed for the SIS model and has Pareto optima, which can be interpreted as the ``best'' vaccination strategies. We provide properties of the corresponding Pareto frontier given by the outcomes (C($η$), L($η$)) of Pareto optimal strategies.

math.OC↗
arXiv2024

An introduction to Galton-Watson trees and their local limits

The aim of this lecture is to give an overview of old and new resultson Bienaymé-Galton-Watson (BGW) trees. After introducing the framework of discretetrees, we first give alternative proofs of classical results on theextinction probability of BGW processes and on thedescription of the processes conditioned on extinction or onnon-extinction. Then, we study recent local limits of critical orsub-critical BGW trees conditioned to be large.

math.PR↗
arXiv2024

Individual based SIS models on (not so) dense large random networks

Starting from a stochastic individual-based description of an SIS epidemic spreading on a random network, we study the dynamics when the size $n$ of the network tends to infinity. We recover in the limit an infinite-dimensional integro-differential equation studied by Delmas, Dronnier and Zitt (2022) for an SIS epidemic propagating on a graphon. Our work covers the case of dense and sparse graphs, provided that the number of edges grows faster than $n$, but not the case of very sparse graphs with $O(n)$ edges. In order to establish our limit theorem, we have to deal with both the convergence of the random graphs to the graphon and the convergence of the stochastic process spreading on top of these random structures: in particular, we propose a coupling between the process of interest and an epidemic that spreads on the complete graph but with a modified infection rate. Keywords: Random graph, mathematical models of epidemics, measure-valued process, large network limit, limit theorem, graphon.

math.PR↗
arXiv2024

Infinite dimensional metapopulation SIS model with generalized incidence rate

We consider an infinite-dimension SIS model introduced by Delmas, Dronnier and Zitt, with a more general incidence rate, and study its equilibria. Unsurprisingly, there exists at least one endemic equilibrium if and only if the basic reproduction number is larger than 1. When the pathogen transmission exhibits one way propagation, it is possible to observe different possible endemic equilibria. We characterize in a general setting all the equilibria, using a decomposition of the space into atoms, given by the transmission operator. We also prove that the proportion of infected individuals converges to an equilibrium, which is uniquely determined by the support of the initial condition.We extend those results to infinite-dimensional SIS models with reservoir or with immigration.

math.AP↗
arXiv2024

Off-the-grid prediction and testing for linear combination of translated features

We consider a model where a signal (discrete or continuous) is observed with an additive Gaussian noise process. The signal is issued from a linear combination of a finite but increasing number of translated features. The features are continuously parameterized by their location and depend on some scale parameter. First, we extend previous prediction results for off-the-grid estimators by taking into account here that the scale parameter may vary. The prediction bounds are analogous, but we improve the minimal distance between two consecutive features locations in order to achieve these bounds. Next, we propose a goodness-of-fit test for the model and give non-asymptotic upper bounds of the testing risk and of the minimax separation rate between two distinguishable signals. In particular, our test encompasses the signal detection framework. We deduce upper bounds on the minimal energy,expressed as the $\ell_2$-norm of the linear coefficients, to successfully detect a signal in presence of noise. The general model considered in this paper is a non-linear extension of the classical high-dimensional regression model. It turns out that,in this framework, our upper bound on the minimax separation rate matches (up to a logarithmic factor) the lower bound on the minimax separation rate for signal detection in the high-dimensional linear model associated to a fixed dictionary of features. We also propose a procedure to test whether the features of the observed signal belong to a given finite collection under the assumption that the linear coefficients may vary, but have prescribed signs under the null hypothesis. A non-asymptotic upper bound on the testing risk is given.We illustrate our results on the spikes deconvolution model with Gaussian features on the real line and with the Dirichlet kernel, frequently used in the compressed sensing literature, on the torus.

math.ST↗
arXiv2024

Branching capacity of a random walk in $\mathbb Z^5$

We are interested in the branching capacity of the range of a random walk in $\mathbb Z^d$.Schapira [28] has recently obtained precise asymptotics in the case $d\ge 6$ and has demonstrated a transition at dimension $d=6$. We study the case $d=5$ and prove that the renormalized branching capacity converges in law to the Brownian snake capacity of the range of a Brownian motion. The main step in the proof relies on studying the intersection probability between the range of a critical Branching random walk and that of a random walk, which is of independent interest.

math.PR↗
arXiv2024

Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary

In this paper we observe a set, possibly a continuum, of signals corrupted by noise. Each signal is a finite mixture of an unknown number of features belonging to a continuous dictionary. The continuous dictionary is parametrized by a real non-linear parameter. We shall assume that the signals share an underlying structure by assuming that each signal has its active features included in a finite and sparse set. We formulate regularized optimization problem to estimate simultaneously the linear coefficients in the mixtures and the non-linear parameters of the features. The optimization problem is composed of a data fidelity term and a $(\ell_1,L^p)$-penalty. We call its solution the Group-Nonlinear-Lasso and provide high probability bounds on the prediction error using certificate functions. Following recent works on the geometry of off-the-grid methods, we show that such functions can be constructed provided the parameters of the active features are pairwise separated by a constant with respect to a Riemannian metric.When the number of signals is finite and the noise is assumed Gaussian, we give refinements of our results for $p=1$ and $p=2$ using tail bounds on suprema of Gaussian and $χ^2$ random processes. When $p=2$, our prediction error reaches the rates obtained by the Group-Lasso estimator in the multi-task linear regression model. Furthermore, for $p=2$ these prediction rates are faster than for $p=1$ when all signals share most of the non-linear parameters.

stat.ML↗
arXiv2024

Branching capacity and Brownian snake capacity

The branching capacity has been introduced by [Zhu 2016] as the limit of the hitting probability of a symmetric branching random walk in $\mathbb Z^d$, $d\ge 5$. Similarly, we define the Brownian snake capacity in $\mathbb R^d$, as the scaling limit of the hitting probability by the Brownian snake starting from afar. Then, we prove our main result on the vague convergence of the rescaled branching capacity towards this Brownian snake capacity. Our proof relies on a precise convergence rate for the approximation of the branching capacity by hitting probabilities.

math.PR↗
arXiv2023

Probability-graphons: Limits of large dense weighted graphs

We introduce probability-graphons which are probability kernels that generalize graphons to the case of weighted graphs. Probability-graphons appear as the limit objects to study sequences of large weighted graphs whose distribution of subgraph sampling converge. The edge-weights are taken from a general Polish space, which also covers the case of decorated graphs. Here, graphs can be either directed or undirected. Starting from a distance $d_m$ inducing the weak topology on measures, we define a cut distance on probability-graphons, making it a Polish space, and study the properties of this cut distance. In particular, we exhibit a tightness criterion for probability-graphons related to relative compactness in the cut distance. We also prove that under some conditions on the distance $d_m$, which are satisfied for some well-know distances like the Prohorov distance, and the Fortet-Mourier and Kantorovitch-Rubinstein norms, the topology induced by the cut distance on the spaceof probability-graphons is independent from the choice of $d_m$. Eventually, we prove that this topology coincides with the topology induced by the convergence in distribution of the sampled subgraphs.

cs.DM↗
arXiv2023

Conditioning Bienaym{é}-Galton-Watson trees to have large sub-populations

We study the local limit in distribution of Bienaym{é}-Galton-Watson trees conditioned on having large sub-populations. Assuming a generic and aperiodic condition on the offspring distribution, we prove the existence of a limit given by a Kesten's tree associated with a certain critical offspring distribution.

math.PR↗
arXiv2023

Atoms and associated spectral properties for positive operators on L^p

Inspired by Schwartz, Jang-Lewis and Victory, who study in particular generalizations of triangularizations of matrices to operators, we shall give for positive operators on Lebesgue spaces equivalent definitions of atoms (maximal irreducible sets). We also characterize positive power compact operators having a unique non-zero atom which appears as a natural generalization of irreducible operators and are also considered in epidemiological models. Using the different characterizations of atoms, we also provide a short proof for the representation of the ascent of a positive power compact operator as the maximal length in the graph of critical atoms.

math.SP↗