We consider p independent Brownian motions in ℝd. We assume that p ≥ 2 and p(d- 2) < d. Let ℓt denote the intersection measure of the p paths by time t, i.e., the random measure on ℝd that assigns to any measurable set A ⊂ ℝd the amount of intersection local time of the motions spent in A by time t. Earlier results of Chen derived the logarithmic asymptotics of the upper tails of the total mass ℓt(ℝd) as t →∞. In this paper, we derive a large-deviation principle for the normalised intersection measure t-pℓt on the set of positive measures on some open bounded set B ⊂ ℝd as t →∞ before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the p motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set U ⊂ B. This extends earlier studies on the intersection measure by König and Mörters.

Cette thèse s'inscrit dans le domaine des matrices aléatoires et des techniques de grandes déviations. On s'attachera dans un premier temps à donner des inégalités de déviations pour différentes fonctionnelles du spectre qui reflètent leurs comportement de grandes déviations, pour des matrices de Wigner vérifiant une propriété de concentration indexée par un paramètre alpha ∈ (0,2]. Nous présenterons ensuite le principe de grandes déviations obtenu pour la plus grande valeur propre des matrices de Wigner sans queues Gaussiennes, dans la lignée du travail de Bordenave et Caputo, puis l'étude des grandes déviations des traces de matrices aléatoires que l'on aborde dans trois cas : le cas des beta-ensembles, celui des matrices de Wigner Gaussiennes, et enfin des matrices de Wigner sans queues Gaussiennes. Le cas Gaussien a été l'occasion de revisiter la preuve de Borell et Ledoux des grandes déviations des chaos de Wiener, que l'on prolonge en proposant un énoncé général de grandes déviations qui nous permet donner une autre preuve des principes de grandes déviations des matrices de Wigner sans queues Gaussiennes. Enfin, nous donnons une nouvelle preuve des grandes déviations de la mesure spectrale empirique des beta-ensembles associés à un potentiel quadratique, qui ne repose que sur leur représentation tridiagonale. / This thesis falls within the theory of random matrices and large deviations techniques. We mainly consider large deviations problems which involve a heavy-tail phenomenon. In a first phase, we will focus on finding concentration inequalities for different spectral functionals which reflect their large deviations behavior, for random Hermitian matrices satisfying a concentration property indexed by some alpha ∈ (0,2]. Then we will present the large deviations principle we obtained for the largest eigenvalue of Wigner matrices without Gaussian tails, in line with the work of Bordenave and Caputo. Another example of heavy-tail phenomenon is given by the large deviations of traces of random matrices which we investigate in three cases: the case of beta-ensembles, of Gaussian Wigner matrices, and the case of Wigner matrices without Gaussian tails. The Gaussian case was the opportunity to revisit Borell and Ledoux's proof of the large deviations of Wiener chaoses, which we investigate further by proposing a general large deviations statement, allowing us to give another proof of the large deviations principles known for the Wigner matrices without Gaussian tail. Finally, we give a new proof of the large deviations principles for the beta-ensembles with a quadratic potential, which relies only on the tridiagonal representation of these models. In particular, this result gives a proof of the large deviations of the GUE and GOE which does not rely on the knowledge of the law of the spectrum.

Nesta dissertação de mestrado, vamos apresentar uma prova para os grandes desvios para variáveis aleatórias independentes e identicamente distribuídas com todos os momentos finitos e para a medida empírica de cadeias de Markov com espaço de estados finito e tempo discreto. Além disso, abordaremos os teoremas de Sanov e Gärtner-Ellis. / In this master thesis it is presented a proof of the large deviations for independent and identically distributed random variables with all finite moments and for the empirical measure of Markov chains with finite state space and with discrete time. Moreover, we address the theorems of Sanov and of Gartner-Ellis.

Nesta dissertação de mestrado, vamos apresentar uma prova para os grandes desvios para variáveis aleatórias independentes e identicamente distribuídas com todos os momentos finitos e para a medida empírica de cadeias de Markov com espaço de estados finito e tempo discreto. Além disso, abordaremos os teoremas de Sanov e Gärtner-Ellis. / In this master thesis it is presented a proof of the large deviations for independent and identically distributed random variables with all finite moments and for the empirical measure of Markov chains with finite state space and with discrete time. Moreover, we address the theorems of Sanov and of Gartner-Ellis.

Orientador: Miguel Natálio Abadi / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T15:39:00Z (GMT). No. of bitstreams: 1 Lambert_Rodrigo_M.pdf: 3549677 bytes, checksum: 663438e1feb8f7092723382b6846bc9c (MD5) Previous issue date: 2010 / Resumo: Seja x um alfabeto finito ou enumerável, e considere o espaço de todas as sequências finitas compostas por concatenação de símbolos desse alfabeto. A essas sequências daremos o nome de palavras. Denotaremos por xn conjunto de todas as palavras de tamanho n. No presente trabalho, consideramos uma função que leva cada palavra de tamanho n em um número inteiro entre 0 e n - 1. Essa função é definida pelo maior tamanho possível de uma sobreposição da palavra com uma cópia dela mesma transladada, e é chamada de função de sobreposição. A ela daremos o nome de Sn. A relevância da função de sobreposição foi colocada em evidência, entre outros casos, na análise estatística da Recorrência de Poincaré, e possui relação explícita com a entropia do processo. Nesse trabalho, provamos a convergência da distribuição da função de sobreposição, quando a sequência _e escolhida de acordo com relação a n variáveis aleatórias independentes e identicamente distribuídas no alfabeto x. Também apresentamos um limitante para a velocidade dessa convergência. Como consequência, mostramos também a convergência da esperança e da variância da função de sobreposição. / Abstract: We consider the set of finite sequencies of length n over a finite or contable alphabet x. We consider the function defined over xn which gives the size of the maximum overlap of a given sequence with a (shifted) copy of itself. That function will be denoted by overlapping function. We prove the convergence of the distribution of this function when the sequence is chosen according to a product measure, with identically distributed marginals. We give a point-wise upper bound for the velocity of this convergence. As a byproduct, we show the convergence of te mean and the variance of the overlapping function. / Mestrado / Probabilidade / Mestre em Estatística

L’objet de cette thèse concerne l’étude asymptotique des processus de branchement sur-critiques en environnement aléatoire, qui sont une généralisation du processus de Galton-Watson, avec une loi de reproduction choisie aléatoirement et de manière i.i.d. suivant les générations. Dans le cas de non extinction, nous démontrons une succession de résultats asymptotiques plus fins que ceux établis dans des travaux antérieurs. Le chapitre 1 est consacré à l’étude de l’écart relatif entre le processus (Zn) normalisé et la loi normale. Nous établissons une borne de type Berry-Esseen ainsi qu’un développement pour des déviations de type Cramér, généralisant ainsi le théorème central limite et le principe des déviations modérées établis précédemment dans la littérature. Le second chapitre concerne l'asymptotique de la distribution du processus (Zn) ainsi que le moment harmonique critique de la variable limite W de la population normalisée. Nous établissons un équivalent de l'asymptotique de la distribution du processus Zn et donnons une caractérisation des constantes via une équation fonctionnelle similaire au cas du processus de Galton-Watson. Dans le cas des processus de branchement en environnement aléatoire, les résultats améliorent l'équivalent asymptotique de la distribution de Zn établi dans des travaux antérieurs sous normalisation logarithmique, sous la condition que chaque individu donne naissance à au moins un individu. Nous déterminons aussi la valeur critique pour l'existence du moment harmonique de W sous des conditions simples d'existence de moments, qui sont bien plus faibles que les hypothèses imposées dans la littérature, et généralisons le résultat à Z_0=k individus initiaux. Le troisième chapitre est consacré à l'étude de l'asymptotique des moments harmoniques d'ordre r>0 de Zn. Nous établissons un équivalent et donnons une expression des constantes. Le résultat met en évidence un phénomène de transition de phase, relié aux transitions de phase des grandes déviations inférieures du processus (Zn). En application de ce résultat, nous établissons un résultat de grandes déviations inférieures pour le processus (Zn) sous des hypothèses plus faibles que celles imposées dans des travaux précédents. Nous améliorons également la vitesse de convergence dans un théorème central limite vérifié par W_n-W, et déterminons l'asymptotique de la probabilité de grandes déviations pour le ratio Zn+1/Z_n. / The purpose of this Ph.D. thesis is the study of branching processes in a random environment, say (Z_n), which are a generalization of the Galton-Watson process, with the reproduction law chosen randomly in each generation in an i.i.d. manner. We consider the case of a supercritical process, assuming the condition that each individual gives birth to at least one child. The first part of this work is devoted to the study of the relative and absolute distance between the normalized process log Z_n and the normal law. We show a Berry-Esseen bound and establish a Cramér type large deviation expansion, which generalize the central limit theorem and the moderate deviation principle established for log Z_n in previous studies.In the second chapter we study the asymptotic of the distribution of Z_n, and the critical value for the existence of harmonic moments of the limit variable W of the normalized population size. We give an equivalent of the asymptotic distribution of Z_n and characterize the constants by a functional relation which is similar to that obtained for a Galton-Watson process. For a branching process in a random environment, our result generalizes the equivalent of the asymptotic distribution of Z_n established in a previous work in a log-scale, under the condition that each individual gives birth to at least one child. We also characterize the critical value for the existence of harmonic moments of the limit variable W under weaker conditions that in previous studies and generalize this result for processes starting with Z_0=k initial individuals. The third chapter is devoted to the study of the asymptotic of the harmonic moments of order r>0 of Z_n. We show the exact decay rate and give an expression of the limiting constants. The result reveals a phase transition phenomenon which is linked to the phase transitions in the lower large deviations established in earlier studies. As an application, we improve a lower large deviation result for the process (Z_n) under weaker hypothesis than those stated in the literature. Moreover, we also improve the rate of convergence in a central limit theorem for W-W_n and give the asymptotic of the large deviation for the ratio Zn+1/Z_n.

Consider an iterated function system (IFS) that does not necessarily contract uniformly, but instead contracts on average after a finite number of iterations. Under some technical assumptions, previous work by Barnsley, Demko, Elton and Geronimo has shown that such an IFS has a unique invariant probability measure, whilst many (such as Peigné, Hennion and Hervé, Guivarc'h and le Page, Santos and Walkden) have shown that (for different function spaces) the transfer operator associated with the IFS is quasi-compact. A result due to Keller and Liverani allows one to deduce whether the transfer operator remains quasi-compact under suitable, small perturbations. The first part of this thesis proves a large deviations result for IFSs that contract on average using skew product transfer operators, a technique used by Broise to prove a similar result for dynamical systems. The remaining chapters introduce a notion of 'coupled IFSs', an analogue of the traditional coupled map lattices where the base, unperturbed behaviour is determined by an underlying dynamical system. We use transfer operators and Keller and Liverani's theorem to prove that quasi-compactness of the transfer operator is preserved for 'product IFSs' under small perturbations and for coupled IFSs. This allows us to prove a central limit theorem with a rate of convergence for the coupled IFS.

Le but de cette thèse est de développer la théorie de Freidlin-Wentzell pour des modèles des épidémies, afin de prédire le temps mis par les perturbations aléatoires pour éteindre une situation endémique "stable". Tout d'abord nous proposons une nouvelle démonstration plus courte par rapport à celle établit récemment (sous une hypothèse un peu différente, mais satisfaite dans tous les exemples de modèles de maladie infectieuses que nous avons à l'esprit) par Kratz et Pardoux (2017) sur le principe de grandes déviations pour les modèles des épidémies. Ensuite nous établissons un principe de grandes déviations pour des EDS poissoniennes réfléchies au bord d'un ouvert suffisamment régulier. Nous établissons aussi un résultat concernant la zone du bord la plus probable par laquelle le processus solution de l'EDS de Poisson va sortir du domaine d'attraction d'un équilibre stable de sa loi des grands nombres limite. Nous terminons cette thèse par la présentation des méthodes "non standard aux différences finis", appropriés pour approcher numériquement les solutions de nos EDOs ainsi que par la résolution d'un problème de contrôle optimal qui permet d'avoir une bonne approximation du temps d'extinction d'un processus d'infection. / In this thesis, we develop the Freidlin-Wentzell theory for the "natural'' Poissonian random perturbations of the above ODE in Epidemic Dynamics (and similarly for models in Ecology or Population Dynamics), in order to predict the time taken by random perturbations to extinguish a "stable" endemic situation. We start by a shorter proof of a recent result of Kratz and Pardoux (under a somewhat different hypothesis which is satisfied in all the cases we have examined so far), which establishes the large deviations principle for epidemic models. Next, we establish the large deviations principle for reflected Poisonian SDE at the boundary of a sufficiently regular open set. Then, we establish the result for the most likely boundary area by which the process will exit the domain of attraction of a stable equilibrium of an ODE. We conclude this thesis with the presentation of the "non - standard finite difference" methods, suitable to approach numerically the solutions of our ODEs as well as the resolution of an optimal control problem which allows to have a good approximation of the time of extinction of an endemic situation.

Stochastic simulation has been widely applied in many domains. More recently, however, the rapid surge of sophisticated problems such as safety evaluation of intelligent systems has posed various challenges to conventional statistical methods. Motivated by these challenges, in this thesis, we develop novel methodologies with theoretical guarantees and numerical applications to tackle them from different perspectives. In particular, our works can be categorized into two areas: (1) rare-event estimation (Chapters 2 to 5) where we develop approaches to estimating the probabilities of rare events via simulation; (2) model calibration (Chapters 6 and 7) where we aim at calibrating the simulation model so that it is close to reality. In Chapter 2, we study rare-event simulation for a class of problems where the target hitting sets of interest are defined via modern machine learning tools such as neural networks and random forests. We investigate an importance sampling scheme that integrates the dominating point machinery in large deviations and sequential mixed integer programming to locate the underlying dominating points. We provide efficiency guarantees and numerical demonstration of our approach. In Chapter 3, we propose a new efficiency criterion for importance sampling, which we call probabilistic efficiency. Conventionally, an estimator is regarded as efficient if its relative error is sufficiently controlled. It is widely known that when a rare-event set contains multiple "important regions" encoded by the dominating points, importance sampling needs to account for all of them via mixing to achieve efficiency. We argue that the traditional analysis recipe could suffer from intrinsic looseness by using relative error as an efficiency criterion. Thus, we propose the new efficiency notion to tighten this gap. In particular, we show that under the standard Gartner-Ellis large deviations regime, an importance sampling that uses only the most significant dominating points is sufficient to attain this efficiency notion. In Chapter 4, we consider the estimation of rare-event probabilities using sample proportions output by crude Monte Carlo. Due to the recent surge of sophisticated rare-event problems, efficiency-guaranteed variance reduction may face implementation challenges, which motivate one to look at naive estimators. In this chapter we construct confidence intervals for the target probability using this naive estimator from various techniques, and then analyze their validity as well as tightness respectively quantified by the coverage probability and relative half-width. In Chapter 5, we propose the use of extreme value analysis, in particular the peak-over-threshold method which is popularly employed for extremal estimation of real datasets, in the simulation setting. More specifically, we view crude Monte Carlo samples as data to fit on a generalized Pareto distribution. We test this idea on several numerical examples. The results show that in the absence of efficient variance reduction schemes, it appears to offer potential benefits to enhance crude Monte Carlo estimates. In Chapter 6, we investigate a framework to develop calibration schemes in parametric settings, which satisfies rigorous frequentist statistical guarantees via a basic notion that we call eligibility set designed to bypass non-identifiability via a set-based estimation. We investigate a feature extraction-then-aggregation approach to construct these sets that target at multivariate outputs. We demonstrate our methodology on several numerical examples, including an application to calibration of a limit order book market simulator. In Chapter 7, we study a methodology to tackle the NASA Langley Uncertainty Quantification Challenge, a model calibration problem under both aleatory and epistemic uncertainties. Our methodology is based on an integration of distributionally robust optimization and importance sampling. The main computation machinery in this integrated methodology amounts to solving sampled linear programs. We present theoretical statistical guarantees of our approach via connections to nonparametric hypothesis testing, and numerical performances including parameter calibration and downstream decision and risk evaluation tasks.

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