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In this paper, we show a law of large numbers relating the bubbles of 1-dimensional, N-parameter Brownian Sheet on a bounded domain and the local time on that domain. This result generalizes the work of Mountford (Brownian sheet, local time and bubbles. In: Seminaire de Probabilites, XXXVII. Lecture Notes in Mathematics, vol 1832. Springer, Berlin, 2003, pp. 19–215).

The Lamperti correspondence gives a prominent role to two random time changes: the exponential functional of a Levy process drifting to \(\infty\) and its inverse, the clock of the corresponding positive self-similar process. We describe here asymptotical properties of these clocks in large time, extending the results of Yor and Zani (Bernoulli 7, 351–362, 2001).

In this paper, we state a large deviation principle (LDP) and sharp LDP for maximum likelihood estimators of drift coefficients of generalized squared radial Ornstein–Uhlenbeck processes. For that purpose, we present an LDP in a class of non-steep cases, where the Gartner–Ellis theorem cannot be applied.

We show a large deviations principle for the family of random variables \(\{ \frac{1}{t}\int_0^t 1 _{B_u } >0du\} \) when t→+∞, where B=(Bu,u≥0) is a standard linear Brownian motion.

We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we thus recover Bott’s periodicity.

We introduce a family of convex (concave) functions called sup (inf) of powers, which are used as generator functions for a special type of quasi-arithmetic means. Using these means, we generalize the large deviation result on self-normalized statistics that was obtained in the homogeneous case by [Q.-M. Shao, Self-normalized large deviations. Ann. Probab. 25 (1997) 285–328]. Furthermore, in the homogenous case, we derive the Bahadur exact slope for tests using self-normalized statistics.

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani’s result [M. Zani, Large deviations for squared radial Ornstein–Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25–42] to the non-steep case will provide the required large deviations principle.

We show the law of large numbers, the central limit theorem and the large-deviation principle for the Bessel clock ∈t0t\rm d}s/(Rs(ν))2, where (Rt(ν), t≥0) is a Bessel process of index ν>0. We also give functional versions of these limit theorems.

This thesis focuses on the study of the $\tau$-coherence of an high-dimensional $(n \times p)$-observation matrix with $p>>n$, where $n$ is the number of individuals and $p$ the number of variables. The $\tau$-coherence is defined as the largest magnitude of the entries of the empirical correlation matrix outside a central band (with a bandwith $\tau$). The first chapter is devoted to the presentation of the Chen-Stein method, which is an approximation of weakly dependent events by a Poisson distribution, and to some bibliography concerning coherence. The second and third chapter focus on the limiting behaviour of $\tau$-coherence in a case where the observations are assumed to be Gaussian with bandwise (resp.blockwise) covariance in Chapter 2 (resp. Chapter 3). In the last chapter, we propose a Monte-Carlo simulation procedure allowing us to study numerically the limiting distribution of the $\tau$-coherence for large (Big Data) matrices. We use a splitting strategy of our matrices and HPC method such as GPGPU computation in order to, from one side, being able to compute correlation matrices even if they are too large to be loaded in a computer, and on the other side, to reduce computation time. Finally the appendix is devoted to some technical results.\\; Cette thèse concerne l'étude de la $\tau$-cohérence d'une matrice d'observations aléatoires de grande taille $(n \times p)$ où $p >> n$ et avec $p$ le nombre de variables observées sur $n$ individus. La $\tau$-cohérence est alors définie comme étant le maximum, en valeur absolue, des coefficients de la matrice de corrélation empirique associée, en dehors d'une bande centrale de largeur $\tau$. Le premier chapitre est consacré à la présentation de la méthode de Chen-Stein qui permet l'approximation d'événements faiblement dépendants par une loi de Poisson et à la présentation des travaux de T. Cai et T. Jiang concernant la cohérence. Les deuxième et troisième chapitres sont consacrés à l'étude du comportement asymptotique de la $\tau$-cohérence dans le cas où les observations proviennent d'un modèle gaussien et où la matrice de covariance possède une structure par bandes (chapitre 2) ou par blocs (chapitre 3). Dans le chapitre 4, nous présentons une méthode de simulation par réplications Monte-Carlo pour étudier numériquement la distribution asymptotique de la $\tau$-cohérence. Nous utilisons des stratégies de découpage de nos matrices et des teThis thesis focuses on the study of the $\tau$-coherence of an high-dimensional $(n \times p)$-observation matrix with $p>>n$, where $n$ is the number of individuals and $p$ the number of variables. The $\tau$-coherence is defined as the largest magnitude of the entries of the empirical correlation matrix outside a central band (with a bandwith $\tau$). The first chapter is devoted to the presentation of the Chen-Stein method, which is an approximation of weakly dependent events by a Poisson distribution, and to some bibliography concerning coherence. The second and third chapter focus on the limiting behaviour of $\tau$-coherence in a case where the observations are assumed to be Gaussian with bandwise (resp.blockwise) covariance in Chapter 2 (resp. Chapter 3). In the last chapter, we propose a Monte-Carlo simulation procedure allowing us to study numerically the limiting distribution of the $\tau$-coherence for large (Big Data) matrices. We use a splitting strategy of our matrices and HPC method such as GPGPU computation in order to, from one side, being able to compute correlation matrices even if they are too large to be loaded in a computer, and on the other side, to reduce computation time. Finally the appendix is devoted to some technical results.\\chniques HPC telles que le calcul en GPU pour, d'une part pouvoir calculer des corrélations sur des matrices trop grandes pour être stockées, et d'autre part réduire le temps de calcul. Nous présentons en annexe de ce manuscrit des éléments de preuves supplémentaires.\\

The Cox-Ingersoll-Ross process and the Heston process are widely used in financial mathematics for pricing and hedging or to model interest rates. In this thesis, we focus on estimating their parameters using continuous-time observations. Firstly, we restrict ourselves to the most tractable situation where the CIR processis geometrically ergodic and does not vanish. We establish a large deviations principle for the maximum likelihood estimator of the couple of dimensionnal and drift parameters of a CIR process. Then we establish a moderate deviations principle for the maximum likelihood estimator of the four parameters of an Heston process, as well as for the maximum likelihood estimator of the couple of parameters of a CIR process. In contrast to the previous literature, parameters are estimated simultaneously. Secondly, we do not restrict ourselves anymore to the case where the CIR process never reaches zero and we introduce a new weighted least squares estimator for the quadruplet of parameters of an Heston process. We establish its strong consitency and asymptotic normality, and we illustrate numerically its good performances.; Les processus de Cox-Ingersoll-Ross et de Heston jouent un rôle prépondérant dans la modélisation mathématique des cours d’actifs financiers ou des taux d’intérêts. Dans cette thèse, on s’intéresse à l’estimation de leurs paramètres à partir de l’observation en temps continu d’une de leurs trajectoires. Dans un premier temps, on se place dans le cas où le processus CIR est géométriquement ergodique et ne s’annule pas. On établit alors un principe de grandes déviationspour l’estimateur du maximum de vraisemblance du couple des paramètres de dimension et de dérive d’un processus CIR. On établit ensuite un principe de déviations modérées pour l’estimateur du maximum de vraisemblance des quatre paramètres d’un processus de Heston, ainsi que pour l’estimateur du maximum de vraisemblance du couple des paramètres d’un processus CIR. Contrairement à ce qui a été fait jusqu’ici dans la littérature,les paramètres sont estimés simultanément. Dans un second temps, on ne se restreint plus au cas où le processus CIR n’atteint jamais zéro et on propose un nouvel estimateur des moindres carrés pondérés pour le quadruplet des paramètres d’un processus de Heston.On établit sa consistance forte et sa normalité asymptotique, et on illustre numériquement ses bonnes performances.