Relations between connected and self-avoiding walks in a digraph

Walks in a directed graph can be given a partially ordered structure that extends to possibly unconnected objects, called hikes. Studying the incidence algebra on this poset reveals unsuspected relations between walks and self-avoiding hikes. These relations are derived by considering truncated versions of the characteristic polynomial of the weighted adjacency matrix, resulting in a collection of matrices whose entries enumerate the self-avoiding hikes of length $\ell$ from one vertex to another

Estimation error for blind Gaussian time series prediction

We tackle the issue of the blind prediction of a Gaussian time series. For this, we construct a projection operator build by plugging an empirical covariance estimation into a Schur complement decomposition of the projector. This operator is then used to compute the predictor. Rates of convergence of the estimates are given

Champs et processus gaussiens indexés par des graphes, estimation et prédiction

L'objet de cette thèse est l'étude de processus gaussiens indexés par des graphes. Le but est de fournir des outils pour la modélisation, l'estimation, et la prédiction de tels champs ou processus, utilisant fortement la structure du graphe. Dans un premier travail, nous nous intéressons au problème de prédiction aveugle de séries chronologiques et montrons que le biais de l'erreur de prédiction décroît à une vitesse qui dépend de la régularité de la densité spectrale, sous une hypothèse de courte mémoire. Nous utilisons ensuite la structure spectrale du graphe pour proposer des modèles de covariance pour des champs gaussiens indexés par ce graphe. Cela fournit immédiatement une représentation spectrale, qui permet d'étendre l'approximation de Whittle et l'estimation par quasi-maximum de vraissemblance à ce cadre. Enfin, cette constructionet le lemmede Szegöpeuventêtre étendus au cas spatiotemporel. Cela permet de mettre en pratique la théorie sur des données réelles.In this work, westudy Gaussian processes indexed by graphs.Weaim at providing tools for modelisation, estimation, and prediction, that uses the structure of the underlying graphs. In the first Chapter,we deal with the blind prediction problem, and compute, in the case of short range dependancy, the rate of convergence of the bias in the prediction error. This rate depends on the regularity of the spectral density of the process. Then, we use the eigenstructure of the adjacency operatorofa graphto propose some models for covariance operators of Gaussian fields indexedby this graph. It leads to aspectral representation for this operator, that can be used to extend Whittle approximation, and quasi-maximum likelihoo destimation. Finally, this construction may be extended to the spatio-temporal case, where the Szegö lemma still holds

Estimating the transition matrix of a Markov chain observed at random times

In this paper we develop a statistical estimation technique to recover the transition kernel $P$ of a Markov chain $X=(X_m)_{m \in \mathbb N}$ in presence of censored data. We consider the situation where only a sub-sequence of $X$ is available and the time gaps between the observations are iid random variables. Under the assumption that neither the time gaps nor their distribution are known, we provide an estimation method which applies when some transitions in the initial Markov chain $X$ are known to be unfeasible. A consistent estimator of $P$ is derived in closed form as a solution of a minimization problem. The asymptotic performance of the estimator is then discussed in theory and through numerical simulations

Gaussian stationary processes over graphs, general frame and maximum likelihood identification

In this paper, using spectral theory of Hilbertian operators, we study ARMA Gaussian processes indexed by graphs. We extend Whittle maximum likelihood estimation of the parameters for the corresponding spectral density and show their asymptotic optimality

Blind forecasting for Gaussian time-series

International audienceLe but de cet exposé est de fournir, dans le cadre des séries chronologiques, un estimateur "aveugle" de l'opérateur de projection sur le passé infini. Il s'agit de donner un prédicteur, lorsque la structure de covariance est inconnue, et qu'un unique échantillon est disponible pour simultanément estimer la covariance et prédire le processus. Nous obtenons la vitesse de convergence en erreur quadratique, en utilisant un résultat de concentration sur la covariance empirique, et une astucieuse décomposition de Schur donnant une forme alternative de ce projecteur. La vitesse est alors obtenue en fonction de la régularité de la densité spectrale

Diffusivity of a random walk on random walks

We consider a random walk $\left(Z^{(1)}_n, \cdots, Z^{(K+1)}_n \right) \in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor $\sigma_K^2 = \frac{2}{K+2}$ with respect to the case of the classical simple random walk without constraint

A combinatorial approach to a model of constrained random walkers

In [1], the authors consider a random walk $(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with the one obtained in [1]