arXiv subjects

Jean-François Delmas

Publications and source records attributed to Jean-François Delmas.

At least 55 records · Page 3Linked to original sources

arXiv2016

Cost functionals for large (uniform and simply generated) random trees

Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of internal nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary second moment computations. We recover results first given by Fill and Kapur (2004) and then by Fill and Janson (2009). In the simply generated case, this relies on the convergence of conditioned Galton-Watson towards stable Lévy trees. We recover results first given by Janson (2003 and 2016) in the quadratic case and give a generalization to the stable case.

math.CO↗

arXiv2016

Local limits of galton-watson trees conditioned on the number of protected nodes

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.

math.PR↗

arXiv2016

Fast adaptive estimation of log-additive exponential models in Kullback-Leibler divergence

We study the problem of nonparametric estimation of density functions with a product form on the domain $\triangle=\{( x_1, \ldots, x_d)\in \mathbb{R}^d, 0\leq x_1\leq \dots \leq x_d \leq 1\}$. Such densities appear in the random truncation model as the joint density function of observations. They are also obtained as maximum entropy distributions of order statistics with given marginals. We propose an estimation method based on the approximation of the logarithm of the density by a carefully chosen family of basis functions. We show that the method achieves a fast convergence rate in probability with respect to the Kullback-Leibler divergence for densities whose logarithm belongs to a Sobolev function class with known regularity. In the case when the regularity is unknown, we propose an estimation procedure using convex aggregation of the log-densities to obtain adaptability. The performance of this method is illustrated in a simulation study.

math.ST↗

arXiv2016

Optimal exponential bounds for aggregation of estimators for the Kullback-Leibler loss

We study the problem of model selection type aggregation with respect to the Kullback-Leibler divergence for various probabilistic models. Rather than considering a convex combination of the initial estimators $f_1, \ldots, f_N$, our aggregation procedures rely on the convex combination of the logarithms of these functions. The first method is designed for probability density estimation as it gives an aggregate estimator that is also a proper density function, whereas the second method concerns spectral density estimation and has no such mass-conserving feature. We select the aggregation weights based on a penalized maximum likelihood criterion. We give sharp oracle inequalities that hold with high probability, with a remainder term that is decomposed into a bias and a variance part. We also show the optimality of the remainder terms by providing the corresponding lower bound results.

math.ST↗

arXiv2015

Maximum entropy distribution of order statistics with given marginals

We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.

math.ST↗

arXiv2014

Total length of the genealogical tree for quadratic stationary continuous-state branching processes

We prove the existence of the total length process for the genealogical tree of a population model with random size given by a quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingma coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage tree in this model.

math.PR↗

arXiv2013

Maximum entropy copula with given diagonal section

We consider copulas with a given diagonal section and compute the explicit density of the unique optimal copula which maximizes the entropy. In this sense, this copula is the least informative among the copulas with a given diagonal section. We give an explicit criterion on the diagonal section for the existence of the optimal copula and give a closed formula for its entropy. We also provide examples for some diagonal sections of usual bivariate copulas and illustrate the differences between them and the maximum entropy copula with the same diagonal section.

math.ST↗

arXiv2013

A population model with non-neutral mutations using branching processes with immigration

We consider a stationary continuous model of random size population with non-neutral mutations using a continuous state branching process with non-homogeneous immigration. We assume the type (or mutation) of the immigrants is random given by a constant mutation rate measure. We determine some genealogical properties of this process such as: distribution of the time to the most recent common ancestor (MRCA), bottleneck effect at the time to the MRCA (which might be drastic for some mutation rate measures), favorable type for the MRCA, asymptotics of the number of ancestors.

math.PR↗

arXiv2013

Record process on the Continuum Random Tree

By considering a continuous pruning procedure on Aldous's Brownian tree, we construct a random variable $Θ$ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned by $n$ leaves.

math.PR↗

arXiv2012

Pruning of CRT-sub-trees

We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the Lévy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate $λ$, see Duquesne and Le Gall (2002). The tree-valued process, as $λ$ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a tree-valued process indexed by two parameters the pruning parameter $θ$ and the intensity $λ$. Our main results are: construction and marginals of the pruning process, representation of the pruning process (forward in time that is as $θ$ increases) and description of the growing process (backward in time that is as $θ$ decreases) and distribution of the ascension time (or explosion time of the backward process) as well as the tree at the ascension time. A by-product of our result is that the super-critical Lévy trees independently introduced by Abraham and Delmas (2012) and Duquesne and Winkel (2007) coincide. This work is also related to the pruning of discrete Galton-Watson trees studied by Abraham, Delmas and He (2012).

math.PR↗

arXiv2012

A construction of a $β$-coalescent via the pruning of Binary Trees

Considering a random binary tree with $n$ labelled leaves, we use a pruning procedure on this tree in order to construct a $β(3/2,1/2)$-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the $β$-coalescent process up to some time change. These two constructions unable us to obtain results on the coalescent process such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

math.PR↗

arXiv2012

A continuum-tree-valued Markov process

We present a construction of a Lévy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $ψ$ be a critical branching mechanism. We set $ψ_θ(\cdot)=ψ(\cdot+θ)-ψ(θ)$. Let $Θ=(θ_\infty,+\infty)$ or $Θ=[θ_\infty,+\infty)$ be the set of values of $θ$ for which $ψ_θ$ is a branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process $(\ct_θ,θ\inΘ)$, such that $\mathcal{T}_θ$ has branching mechanism $ψ_θ$. It is sub-critical if $θ>0$ and super-critical if $θ<0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $θ$ for which $\mathcal{T}_θ$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT.

math.PR↗

arXiv2012

Exit times for an increasing Lévy tree-valued process

We give an explicit construction of the increasing tree-valued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the tree grows higher than a given height. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree-valued process.

math.PR↗

arXiv2012

Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Lévy processes.

math.PR↗

arXiv2010

Smaller population size at the MRCA time for stationary branching processes

We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distribution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before the most recent common ancestor (MRCA). In particular we show a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of old families which corresponds to the number of individuals involved in the last coalescent event of the genealogical tree. By studying more precisely the genealogical structure of the population, we get asymptotics for the number of ancestors just before the current time. We give explicit computations in the case of the quadratic branching mechanism. In this case, the size of the population at the MRCA is, in mean, less by 1/3 than size of the current population size. We also provide in this case the fluctuations for the renormalized number of ancestors.

math.PR↗

arXiv2009

Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations

We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams' decomposition of the genealogy of the total population given by a continuous random tree, according to the ancestral lineage of the last individual alive. This allows us give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.

math.PR↗

arXiv2009

Does waste-recycling really improve Metropolis-Hastings Monte Carlo algorithm?

The Metropolis Hastings algorithm and its multi-proposal extensions are aimed at the computation of the expectation $<π,f>$ of a function $f$ under a probability measure $π$ difficult to simulate. They consist in constructing by an appropriate acceptation/rejection procedure a Markov chain $(X_k,k\geq 0)$ with transition matrix $P$ such that $π$ is reversible with respect to $P$ and in estimating $<π,f>$ by the empirical mean $I_n(f)=\inv{n}\sum_{k=1}^n f(X_k)$. The waste-recycling Monte Carlo (WR) algorithm introduced by physicists is a modification of the Metropolis-Hastings algorithm, which makes use of all the proposals in the empirical mean, whereas the standard Metropolis-Hastings algorithm only uses the accepted proposals. In this paper, we extend the WR algorithm into a general control variate technique and exhibit the optimal choice of the control variate in terms of asymptotic variance. We also give an example which shows that in contradiction to the intuition of physicists, the WR algorithm can have an asymptotic variance larger than the one of the Metropolis-Hastings algorithm. However, in the particular case of the Metropolis-Hastings algorithm called Boltzmann algorithm, we prove that the WR algorithm is asymptotically better than the Metropolis-Hastings algorithm.

math.PR↗

arXiv2008

Detection of cellular aging in a Galton-Watson process

We consider the bifurcating Markov chain model introduced by Guyon to detect cellular aging from cell lineage. To take into account the possibility for a cell to die, we use an underlying Galton-Watson process to describe the evolution of the cell lineage. We give in this more general framework a weak law of large number, an invariance principle and thus fluctuation results for the average over one generation or up to one generation. We also prove the fluctuations over each generation are independent. Then we present the natural modifications of the tests given by Guyon in cellular aging detection within the particular case of the auto-regressive model.

math.PR↗