arXiv subjects

Ronan Le Guével

Publications and source records attributed to Ronan Le Guével.

11 recordsLinked to original sources

arXiv2022

Feller and ergodic properties of jump-move processes with applications to interacting particle systems

We consider Markov processes that alternate continuous motions and jumps in a general locally compact polish space. Starting from a mechanistic construction, a first contribution of this article is to provide conditions on the dynamics so that the associated transition kernel forms a Feller semigroup, and to deduce the corresponding infinitesimal generator. In a second contribution, we investigate the ergodic properties in the special case where the jumps consist of births and deaths, a situation observed in several applications including epidemiology, ecology and microbiology. Based on a coupling argument, we obtain conditions for the convergence to a stationary measure with a geometric rate of convergence. Throughout the article, we illustrate our results by general examples of systems of interacting particles in R d with births and deaths. We show that in some cases the stationary measure can be made explicit and corresponds to a Gibbs measure on a compact subset of R d. Our examples include in particular Gibbs measure associated to repulsive Lennard-Jones potentials and to Riesz potentials.

math.PR↗

arXiv2021

Optimal L2 -approximation of occupation and local times for symmetric stable processes

The L2-approximation of occupation and local times of a symmetric $α$-stable L{é}vy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when 0 < $α$ $\le$ 1, but only rate optimal for 1 < $α$ $\le$ 2. For this, the exact convergence of the L2-approximation error is proven with explicit constants.

math.PR↗

arXiv2021

Spatial birth-death-move processes : basic properties and estimation of their intensity functions

Many spatio-temporal data record the time of birth and death of individuals, along with their spatial trajectories during their lifetime, whether through continuous-time observations or discrete-time observations. Natural applications include epidemiology, individual-based modelling in ecology, spatio-temporal dynamics observed in bio-imaging, and computer vision. The aim of this article is to estimate in this context the birth and death intensity functions, that depend in full generality on the current spatial configuration of all alive individuals. While the temporal evolution of the population size is a simple birth-death process, observing the lifetime and trajectories of all individuals calls for a new paradigm. To formalise this framework, we introduce spatial birth-death-move processes, where the birth and death dynamics depends on the current spatial configuration of the population and where individuals can move during their lifetime according to a continuous Markov process with possible interactions.We consider non-parametric kernel estimators of their birth and death intensity functions. The setting is original because each observation in time belongs to a non-vectorial, infinite dimensional space and the dependence between observations is barely tractable. We prove the consistency of the estimators in presence of continuous-time and discrete-time observations, under fairly simple conditions. We moreover discuss how we can take advantage in practice of structural assumptions made on the intensity functions and we explain how data-driven bandwidth selection can be conducted, despite the unknown (and sometimes undefined) second order moments of the estimators. We finally apply our statistical method to the analysis of the spatio-temporal dynamics of proteins involved in exocytosis in cells, providing new insights on this complex mechanism.

math.PR↗

arXiv2021

Minimax and adaptive tests for detecting abrupt and possibly transitory changes in a Poisson process

Motivated by applications in cybersecurity and epidemiology, we consider the problem of detecting an abrupt change in the intensity of a Poisson process, characterised by a jump (non transitory change) or a bump (transitory change) from constant. We propose a complete study from the nonasymptotic minimax testing point of view, when the constant baseline intensity is known or unknown. The question of minimax adaptation with respect to each parameter (height, location, length) of the change is tackled, leading to a comprehensive overview of the various minimax separation rate regimes. We exhibit three such regimes and identify the factors of the two phase transitions, by giving the cost of adaptation to each parameter. For each alternative hypothesis, depending on the knowledge or not of each change parameter, we propose minimax or minimax adaptive tests based on linear statistics, close to CUSUM statistics, or quadratic statistics more adapted to the L 2-distance considered in our minimax criteria and typically more powerful in practice, as our simulation study shows. When the change location or length is unknown, our adaptive tests are constructed from a scan aggregation principle combined with Bonferroni or min-p level correction, and a conditioning trick when the baseline intensity is unknown.

math.ST↗

arXiv2016

The Hausdorff dimension of the range of the Lévy multistable processes

We compute the Hausdorff dimension of the image X(E) of a non random Borel set E $\subset$ [0, 1], where X is a Lévy multistable process in R. This extends the case where X is a classical stable Lévy process by letting the stability exponent $α$ be a smooth function, which leads to non-homogeneous processes because their increments are not stationary and not necessarily independent. Contrary to the situation where the stability parameter is a constant, the dimension depends on the version of the multistable Lévy motion when the process has an infinite first moment.

math.PR↗

arXiv2014

Hausdorff, Large Deviation and Legendre Multifractal Spectra of Lévy Multistable Processes

We compute the Hausdorff multifractal spectrum of two versions of multistable L{é}vy motions. These processes extend classical L{é}vy motion by letting the stability exponent $α$ evolve in time. The spectra provide a decomposition of [0, 1] into an uncountable disjoint union of sets with Hausdorff dimension one. We also compute the increments-based large deviations multifractal spectrum of the independent in-crements multistable L{é}vy motion. This spectrum turns out to be concave and thus coincides with the Legendre multifractal spectrum, but it is different from the Haus-dorff multifractal spectrum. The independent increments multistable L{é}vy motion thus provides an example where the strong multifractal formalism does not hold.

math.PR↗

arXiv2013

On two multistable extensions of stable Lévy motion and their semimartingale representation

We compare two definitions of multistable Lévy motions. Such processes are extensions of classical Lévy motion where the stability index is allowed to vary in time. We show that the two multistable Lévy motions have distinct properties: in particular, one is a pure-jump Markov process, while the other one satisfies neither of these properties. We prove that both are semimartingales and provide semimartingale decompositions.

math.PR↗

arXiv2012

An estimation of the stability and the localisability functions of multistable processes

Multistable processes are tangent at each point to a stable process, but where the index of stability and the index of localisability varies along the path. In this work, we give two estimators of the stability and the localisability functions, and we prove the consistency of those two estimators. We illustrate these convergences with two classical examples, the Levy multistable process and the Linear Multifractional Multistable Motion.

math.PR↗

arXiv2010

Incremental moments and Hölder exponents of multifractional multistable processes

Multistable processes, that is, processes which are, at each "time", tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise exponent is, as expected, lower than the localisability index.

math.PR↗

arXiv2009

A Ferguson-Klass-LePage series representation of multistable multifractional processes and related processes

The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. We present here a construction of multifractional multistable processes, based on the Ferguson-Klass-LePage series representation of stable processes. We consider various particular cases of interest, including multistable Lévy motion, multistable reverse Ornstein-Uhlenbeck process, log-fractional multistable motion and linear multistable multifractional motion. We also compute the finite dimensional distributions of those processes. Finally, we display numerical experiments showing graphs of synthesized paths of such processes.

math.PR↗

arXiv2009

Localisable moving average stable and multistable processes

We study a particular class of moving average processes which possess a property called localisability. This means that, at any given point, they admit a ``tangent process'', in a suitable sense. We give general conditions on the kernel g defining the moving average which ensures that the process is localisable and we characterize the nature of the associated tangent processes. Examples include the reverse Ornstein-Uhlenbeck process and the multistable reverse Ornstein-Uhlenbeck process. In the latter case, the tangent process is, at each time t, a Lévy stable motion with stability index possibly varying with t. We also consider the problem of path synthesis, for which we give both theoretical results and numerical simulations.

math.PR↗