arXiv subjects

Loïc Chaumont

Publications and source records attributed to Loïc Chaumont.

At least 19 recordsLinked to original sources

arXiv2025

Explosion speed of continuous state branching processes indexed by the Esscher transform

A branching process $Z$ is said to be non conservative if it hits $\infty$ in a finite time with positive probability. It is well known that this happens if and only if the branching mechanism $φ$ of $Z$ satisfies $\int_{0+}dλ/|φ(λ)|<\infty$. We construct on the same probability space a family of conservative continuous state branching processes $Z^{(\varepsilon)}$, $\varepsilon\ge0$, each process $Z^{(\varepsilon)}$ having $φ^{(\varepsilon)}(λ)=φ(λ+\varepsilon)-φ(\varepsilon)$ as branching mechanism, and such that the family $Z^{(\varepsilon)}$, $\varepsilon\ge0$ converges a.s.~to $Z$, as $\varepsilon\rightarrow0$. Then we study the speed of convergence of $Z^{(\varepsilon)}$, when $\varepsilon\rightarrow0$, referred to here as the explosion speed. More specifically, we characterize the functions $f$ with $\lim_{\varepsilon\rightarrow0} f(\varepsilon)=\infty$ and such that the first passage times $σ_\varepsilon=\inf\{t:Z^{(\varepsilon)}_t\ge f(\varepsilon)\}$ converge toward the explosion time of $Z$. Necessary and sufficient conditions are obtained for the weak convergence and convergence in $L^1$. Then we give a sufficient condition for the almost sure convergence.

math.PR↗

arXiv2024

Lévy processes resurrected in the positive half-line

A Lévy processes resurrected in the positive half-line is a Markov process obtained by removing successively all jumps that make it negative. A natural question, given this construction, is whether the resulting process is absorbed at 0 or not. We first describe the law of the resurrected process in terms of that of the initial Lévy process. Then in many important classes of Lévy processes, we give conditions for absorption and conditions for non absorption bearing on the characteristics of the initial Lévy process.

math.PR↗

arXiv2022

Creeping of Lévy processes through curves

A Lévy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph $\{(t,f(t)):t\ge0\}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real Lévy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of Lévy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.

math.PR↗

arXiv2021

A lifetime of excursions through random walks and Lévy processes

In celebration of Professor Ron Doney's 80th birthday, we provide a summary of his academic career and contributions to probability theory, as one of the UK's leading probabilists for over 50 years. A version of this note also serves as an introductory article to the volume in honor of Ron Doney [Birkhäuser, 2021], which includes an additional 17 research papers produced by Ron's colleagues and friends.

math.PR↗

arXiv2021

Extinction times of multitype, continuous-state branching processes

A multitype continuous-state branching process (MCSBP) ${\rm Z}=({\rm Z}_{t})_{t\geq 0}$, is a Markov process with values in $[0,\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\mathbb{R}^d$-valued spectrally positive Lévy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive Lévy fields recently obtained in \cite{cma1} and an extension of the Lamperti representation in higher dimension proved in \cite{cpgub}.

math.PR↗

arXiv2019

Fluctuation theory for spectrally positive additive Lévy fields

A spectrally positive additive Lévy field is a multidimensional field obtained as the sum $\mathbf{X}_{\rm t}={\rm X}^{(1)}_{t_1}+{\rm X}^{(2)}_{t_2}+\dots+{\rm X}^{(d)}_{t_d}$, ${\rm t}=(t_1,\dots,t_d)\in\mathbb{R}_+^d$, where ${\rm X}^{(j)}={}^t (X^{1,j},\dots,X^{d,j})$, $j=1,\dots,d$, are $d$ independent $\mathbb{R}^d$-valued Lévy processes issued from 0, such that $X^{i,j}$ is non decreasing for $i\neq j$ and $X^{j,j}$ is spectrally positive. It can also be expressed as $\mathbf{X}_{\rm t}=\mathbb{X}_{\rm t}{\bf 1}$, where ${\bf 1}={}^t(1,1,\dots,1)$ and $\mathbb{X}_{\rm t}=(X^{i,j}_{t_j})_{1\leq i,j\leq d}$. The main interest of spaLf's lies in the Lamperti representation of multitype continuous state branching processes. In this work, we study the law of the first passage times $\mathbf{T}_{\rm r}$ of such fields at levels $-{\rm r}$, where ${\rm r}\in\mathbb{R}_+^d$. We prove that the field $\{(\mathbf{T}_{\rm r},\mathbb{X}_{\mathbf{T}_{\rm r}}),{\rm r}\in\mathbb{R}_+^d\}$ has stationary and independent increments and we describe its law in terms of this of the spaLf $\mathbf{X}$. In particular, the Laplace exponent of $(\mathbf{T}_{\rm r},\mathbb{X}_{\mathbf{T}_{\rm r}})$ solves a functional equation leaded by the Laplace exponent of $\mathbf{X}$. This equation extends in higher dimension a classical fluctuation identity satisfied by the Laplace exponents of the ladder processes. Then we give an expression of the distribution of $\{(\mathbf{T}_{\rm r},\mathbb{X}_{\mathbf{T}_{\rm r}}),{\rm r}\in\mathbb{R}_+^d\}$ in terms of the distribution of $\{\mathbb{X}_{\rm t},{\rm t}\in\mathbb{R}_+^d\}$ by the means of a Kemperman-type formula, well-known for spectrally positive Lévy processes.

math.PR↗

arXiv2019

Density behaviour related to Lévy processes

Let $p_t(x)$, $f_t(x)$ and $q_t^*(x)$ be the densities at time $t$ of a real Lévy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of $p_t(x)$, $f_t(x)$ and $q_t^*(x)$, when $t$ is small and $x$ is large. Then for large $x$, these asymptotic behaviours are compared to this of the density of the Lévy measure. We show in particular that, under mild conditions, if $p_t(x)$ is comparable to $tν(x)$, as $t\rightarrow0$ and $x\rightarrow\infty$, then so is $f_t(x)$.

math.PR↗

arXiv2019

On Doney's striking factorization of the arc-sine law

R. Doney identifies a striking factorization of the arc-sine law in terms of the suprema of two independent stable processes of the same index by an elegant random walks approximation. In this paper, we provide an alternative proof and a generalization of this factorization based on the theory recently developed for the exponential functional of Lévy processes. As a by-product, we provide some interesting distributional properties for these variables and also some new examples of the factorization of the arc-sine law.

math.PR↗

arXiv2019

The entrance law of the excursion measure of the reflected process for some classes of Lévy processes

We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

math.PR↗

arXiv2018

On $\mathbb{R}^d$-valued multi-self-similar Markov processes

An $\mathbb{R}^d$-valued Markov process $X^{(x)}_t=(X^{1,x_1}_t,\dots,X^{d,x_d}_t)$, $t\ge0,x\in\mathbb{R}^d$ is said to be multi-self-similar with index $(α_1,\dots,α_d)\in[0,\infty)^d$ if the identity in law \[(c_iX_t^{i,x_i/c_i};i=1,\dots,d)_{t\ge0}\ed(X_{ct}^{(x)})_{t\ge0}\,,\] where $c=\prod_{i=1}^dc_i^{α_i}$, is satisfied for all $c_1,\dots,c_d>0$ and all starting point $x$. Multi-self-similar Markov processes were introduced by Jacobsen and Yor \cite{jy} in the aim of extending the Lamperti transformation of positive self-similar Markov processes to $\mathbb{R}^d_+$-valued processes. This paper aims at giving a complete description of all $\mathbb{R}^d$-valued multi-self-similar Markov processes. We show that their state space is always a union of open orthants with 0 as the only absorbing state and that there is no finite entrance law at 0 for these processes. We give conditions for these processes to satisfy the Feller property. Then we show that a Lamperti-type representation is also valid for $\mathbb{R}^d$-valued multi-self-similar Markov processes. In particular, we obtain a one-to-one relationship between this set of processes and the set of Markov additive processes with values in $\{-1,1\}^d\times\mathbb{R}^d$. We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes.

math.PR↗

arXiv2018

Short proofs in extrema of spectrally one sided Lévy processes

We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall's identity for spectrally positive Lévy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall's identity, Theorem 2.4 in \cite{mpp} which gives an expression for the law of the supremum of spectrally positive Lévy processes. An analogous formula is obtained for the supremum of spectrally negative Lévy processes.

math.PR↗

arXiv2018

Space and time inversions of stochastic processes and Kelvin transform

Let $X$ be a standard Markov process. We prove that a space inversion property of $X$ implies the existence of a Kelvin transform of $X$-harmonic, excessive and operator-harmonic functions and that the inversion property is inherited by Doob $h$-transforms. We determine new classes of processes having space inversion properties amongst transient processes {satisfying the} time inversion property. {For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly.} We treat in details the examples of free scaled power Bessel processes, non-colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.

math.PR↗

arXiv2017

On distributions determined by their upward, space-time Wiener-Hopf factor

According to the Wiener-Hopf factorization, the characteristic function $φ$ of any probability distribution $μ$ on $\mathbb{R}$ can be decomposed in a unique way as \[1-sφ(t)=[1-χ_-(s,it)][1-χ_+(s,it)]\,,\;\;\;|s|\le1,\,t\in\mathbb{R}\,,\] where $χ_-(e^{iu},it)$ and $χ_+(e^{iu},it)$ are the characteristic functions of possibly defective distributions in $\mathbb{Z}_+\times(-\infty,0)$ and $\mathbb{Z}_+\times[0,\infty)$, respectively. We prove that $μ$ can be characterized by the sole data of the upward factor $χ_+(s,it)$, $s\in[0,1)$, $t\in\mathbb{R}$ in many cases including the cases where: 1) $μ$ has some exponential moments; 2) the function $t\mapstoμ(t,\infty)$ is completely monotone on $(0,\infty)$; 3) the density of $μ$ on $[0,\infty)$ admits an analytic continuation on $\mathbb{R}$. We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: {\it Any probability measure $μ$ on $\mathbb{R}$ whose support is not included in $(-\infty,0)$ is determined by its convolution powers $μ^{*n}$, $n\ge1$ restricted to $[0,\infty)$}. We show that in many instances, the sole knowledge of $μ$ and $μ^{*2}$ restricted to $[0,\infty)$ is actually sufficient to determine $μ$. Then we investigate the analogous problem in the framework of infinitely divisible distributions.

math.PR↗

arXiv2016

Inversion, duality and Doob $h$-transforms for self-similar Markov processes

We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov process $X$, with index $α>0$ can be represented as a path transformation of some Markov additive process (MAP) $(θ,ξ)$ in $S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti transformation. Let us denote by $\widehat{X}$ the self-similar Markov process which is obtained from the MAP $(θ,-ξ)$ through this extended Lamperti transformation. Then we prove that $\widehat{X}$ is in weak duality with $X$, with respect to the measure $π(x/\|x\|)\|x\|^{α-d}dx$, if and only if $(θ,ξ)$ is reversible with respect to the measure $π(ds)dx$, where $π(ds)$ is some $σ$-finite measure on $S_{d-1}$ and $dx$ is the Lebesgue measure on $\mathbb{R}$. Besides, the dual process $\widehat{X}$ has the same law as the inversion $(X_{γ_t}/\|X_{γ_t}\|^2,t\ge0)$ of $X$, where $γ_t$ is the inverse of $t\mapsto\int_0^t\|X\|_s^{-2α}\,ds$. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.

math.PR↗

arXiv2015

On mutations in the branching model for multitype populations

The forest of mutations associated to a multitype branching forest is obtained by merging together all vertices of its clusters and by preserving connections between them. We first show that the forest of mutations of any mulitype branching forest is itself a branching forest. Then we give its progeny distribution and describe some of its crucial properties in terms the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to infinity and when the number of roots tends to infinity. The continuous time case is then investigated by considering multitype branching forests with edge lengths. When mutations are non reversible, we give a representation of their emergence times which allows us to describe the asymptotic behaviour of the latters, when the ratios of successive mutation rates tend to 0.

math.PR↗

arXiv2015

Reconstructing pedigrees using probabilistic analysis of ISSR amplification

Data obtained from ISSR amplification may readily be extracted but only allows us to know, for each gene, if a specific allele is present or not. From this partial information we provide a probabilistic method to reconstruct the pedigree corresponding to some families of diploid cultivars. This method consists in determining for each individual what is the most likely couple of parent pair amongst all older individuals, according to some probability measure. The construction of this measure bears on the fact that the probability to observe the specific alleles in the child, given the status of the parents does not depend on the generation and is the same for each gene. This assumption is then justified from a convergence result of gene frequencies which is proved here. Our reconstruction method is applied to a family of 85 living accessions representing the common broom {\it Cytisus scoparius}.

q-bio.PE↗

arXiv2014

Sustainable deployment of QTLs conferring quantitative resistance to crops: first lessons from a stochastic model

Quantitative plant disease resistance is believed to be more durable than qualitative resistance, since it exerts less selective pressure on the pathogens. However, the process of progressive pathogen adaptation to quantitative resistance is poorly understood, which makes it difficult to predict its durability or to derive principles for its sustainable deployment. Here, we study the dynamics of pathogen adaptation in response to quantitative plant resistance affecting pathogen reproduction rate and its carrying capacity. We developed a stochastic model for the continuous evolution of a pathogen population within a quantitatively resistant host. We assumed that pathogen can adapt to a host by the progressive restoration of reproduction rate or of carrying capacity, or of both. Our model suggests that a combination of QTLs affecting distinct pathogen traits was more durable if the evolution of repressed traits was antagonistic. Otherwise, quantitative resistance that depressed only pathogen reproduction was more durable. In order to decelerate the progressive pathogen adaptation, QTLs that decrease the pathogen's ability to extend must be combined with QTLs that decrease the spore production per lesion or the infection efficiency or that increase the latent period. Our theoretical framework can help breeders to develop principles for sustainable deployment of quantitative trait loci..

q-bio.PE↗

arXiv2014

Breadth first search coding of multitype forests with application to Lamperti representation

We obtain a bijection between some set of multidimensional sequences and this of $d$-type plane forests which is based on the breadth first search algorithm. This coding sequence is related to the sequence of population sizes indexed by the generations, through a Lamperti type transformation. The same transformation in then obtained in continuous time for multitype branching processes with discrete values. We show that any such process can be obtained from a $d^2$ dimensional compound Poisson process time changed by some integral functional. Our proof bears on the discretisation of branching forests with edge lengths.

math.PR↗