In this thesis we study a scalar uniform and non-uniform quantization of random processes (or signals) in average case setting. Quantization (or discretization) of a signal is a standard task in all nalog/digital devices (e.g., digital recorders, remote sensors etc.). We evaluate the necessary memory capacity (or quantization rate) needed for quantized process realizations by exploiting the correlation structure of the model random process. The thesis consists of an introductory survey of the subject and related theory followed by four included papers (A-D). In Paper A we develop a quantization coding method when quantization levels crossings by a process realization are used for its coding. Asymptotical behavior of mean quantization rate is investigated in terms of the correlation structure of the original process. For uniform and non-uniform quantization, we assume that the quantization cellwidth tends to zero and the number of quantization levels tends to infinity, respectively. In Papers B and C we focus on an additive noise model for a quantized random process. Stochastic structures of asymptotic quantization errors are derived for some bounded and unbounded non-uniform quantizers when the number of quantization levels tends to infinity. The obtained results can be applied, for instance, to some optimization design problems for quantization levels. Random signals are quantized at sampling points with further compression. In Paper D the concern is statistical inference for run-length encoding (RLE) method, one of the compression techniques, applied to quantized stationary Gaussian sequences. This compression method is widely used, for instance, in digital signal and image processing. First, we deal with mean RLE quantization rates for various probabilistic models. For a time series with unknown stochastic structure, we investigate asymptotic properties (e.g., asymptotic normality) of two estimates for the mean RLE quantization rate based on an observed sample when the sample size tends to infinity. These results can be used in communication theory, signal processing, coding, and compression applications. Some examples and numerical experiments demonstrating applications of the obtained results for synthetic and real data are presented.

Le but à long terme de mon travail de recherche est de trouver une alternative viable à la méthode conforme, qui nous permettrait de mieux comprendre la structure géométrique de l'espace des solutions des équations de contrainte. L'avantage du modèle de Maxwell (the drift model) par rapport aux modèles plus classiques est la présence des paramètres supplémentaires. Le prix à payer, par contre, sera que la complexité analytique du système correspondant. Ma thèse a été structuré en deux parties : a. Existence sous la condition de petitesse des données initiales. Nous avons montré que le système de Maxwell est raisonnable dans le sens où nous pouvons le résoudre, malgré sa forte nonliniarité, sous des conditions de petitesse sur ses coefficients, en dimension 3, 4 et 5. Par conséquent, l'ensemble des solutions est non-vide. b. Stabilité Nous montrons la stabilité des solutions du système: ce résultat est obtenu en dimension 3,4 et 5, dans le cas où la métrique est conformément plate, et le drift et petit / The long-term goal of my work is to find a viable alternative to the conformal method, which would allow us to better understand the geometry of the space of solutions of the constraint equations. The advantage of Maxwell's model (the drift model) is the presence of additional parameters. Its downside, however, is that it proves to be much more difficult from an analytic standpoint. My thesis is structued in two parts: a. Existence under suitable smallness conditions. We show that Maxwell's system is sufficiently reasonable: it can be solved even given the presence of focusing non linearities. We prove this under smallness conditions of its coefficients, and in dimensions 3,4 and 5. An immediate consequence is that the set of solutions is non-empty. b. Stability. We verify that the solutions of the system are stable: this result holds in dimensions 3,4 and 5, when the metric is conformally flat and the drift is small

Indiana University-Purdue University Indianapolis (IUPUI) / We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_)$ for all values of the real parameter $\gamma$.

Ce mémoire de thèse est consacré à l’équation de Fokker-Planckpartial_ f=∆f+div(Ef).Il est subdivisé en deux parties :une partie linéaire et une partie non linéaire. Dans la partie linéaire on considère un champ de vecteur E(x) dépendant seulement de x. Cette partie est constituée des chapitres 3, 4 et 5. Dans le chapitre 3 on montre que l’opérateur linéaire Lf :=∆ f + div(E f ) est le générateur d’un semi-groupe fortement continu (SL(t))_{t≥0} dans tous les espaces L^p. On y établit également que le semi-groupe (SL(t))_{t≥0} est positif et ultracontractif. Dans le chapitre 4 nous montrons comment est qu’une décomposition adéquate de l’opérateur L permet d’établir certaines propriétés du semi-groupe (SL(t))_{t≥0} notamment sa bornitude. Le chapitre 5 est consacré à l’existence d’un état d’équilibre. De plus on y montre que cet état d’équi- libre est asymptotiquement stable. Dans la partie non linéaire on considère un champ de vecteur de la forme E(x,f) := x+nabla (a*f) ou a et f sont des fonctions assez régulières et * est l’opérateur de convolution. Cette parties est contituée des chapitre 6 et 7. Dans le chapitre 6 nous établissons que poura appartenant à W^{2,infini}_locl’équation de Fokker-Planck non linéaire admet une unique solution locale dans l’espace L^2_{K_alpha} (R^d). Dans le dernier chapitre nous montrons que le problème non linéaire admet une solution globale. De plus cette solution dépend continument des données. / This thesis is devoted to the Fokker-Planck équation partial_t f =∆f + div(E f).It is divided into two parts. The rst part deals with the linear problem. In this part we consider a vector E(x) depending only on x. It is composed of chapters 3, 4 and 5. In chapter 3 we prove that the linear operator Lf :=∆f + div(Ef ) is an in nitesimal generator of a strong continuous semigroup (SL(t))_{t≥0}. We establish also that (SL(t))_{t≥0} is positive and ultracontractive. In chapter 4 we show how an adequate decomposition of the linear operator L allows us to deduce interesting properties for the semigroup (SL(t))_{t≥0}. Indeed using this decomposition we prove that (SL(t))_{t≥0} is a bounded semigroup. In the last chapter of this part we establish that the linear Fokker-Planck admits a unique steady state. Moreover this stationary solution is asymptotically stable.In the nonlinear part we consider a vector eld of the form E(x, f ) := x +nabla (a *f ), where a and f are regular functions. It is composed of two chapters. In chapter 6 we establish that fora in W^{2,infini}_locthe nonlinear problem has a unique local solution in L^2_{K_alpha}(R^d); . To end this part we prove in chapter 7 that the nonlinear problem has a unique global solution in L^2_k(R^d). This solution depends continuously on the data.