Title: Some topics of topological measure theory with application in stochastic analysis Author: Pavel Kříž Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Josef Štěpán, DrSc., Department of Probability and Mathematical Statistics Abstract: This work studies identifications of values of probability limits based on trajectories of convergent (random) sequences. The key concept is the so called Probability Limit Identification Function (PLIF). The main concern is focused on the existence of PLIFs, mainly those, which are measurable and adapted. We also study in more detail special cases, when the convergence in probability and the convergence almost surely coincide. Furthermore, possible applications of the PLIF concept in stochastic analysis (path-wise representations of stochastic integrals and weak solutions of the stochastic differential equations), as well as in estimation theory (the existence of strongly consistent estimators) are outlined. The achieved results are based on analyses of the topologies on spaces of measures, spaces of random variables and spaces of real-valued functions. Keywords: Probability Limit, Identification, Almost-sure Convergence 1

This thesis is concerned with identifying the best decision among a set of possible decisions in the presence of uncertainty. We are primarily interested in situations where the objective function value at any feasible solution needs to be estimated, for example via a ``black-box' simulation procedure. We develop adaptive random search methods for solving such simulation optimization problems. The methods are adaptive in the sense that they use information gathered during previous iterations to decide how simulation effort is expended in the current iteration. We consider random search because such methods assume very little about the structure of the underlying problem, and hence can be applied to solve complex simulation optimization problems with little expertise required from an end-user. Consequently, such methods are suitable for inclusion in simulation software. We first identify desirable features that algorithms for discrete simulation optimization need to possess to exhibit attractive empirical performance. Our approach emphasizes maintaining an appropriate balance between exploration, exploitation, and estimation. We also present two new and almost surely convergent random search methods that possess these desirable features and demonstrate their empirical attractiveness. Second, we develop two frameworks for designing adaptive and almost surely convergent random search methods for discrete simulation optimization. Our frameworks involve averaging, in that all decisions that require estimates of the objective function values at various feasible solutions are based on the averages of all observations collected at these solutions so far. We present two new and almost surely convergent variants of simulated annealing and demonstrate the empirical effectiveness of averaging and adaptivity in the context of simulated annealing. Finally, we present three random search methods for solving simulation optimization problems with uncountable feasible regions. One of the approaches is adaptive, while the other two are based on pure random search. We provide conditions under which the three methods are convergent, both in probability and almost surely. Lastly, we include a computational study that demonstrates the effectiveness of the methods when compared to some other approaches available in the literature.

Les processus autorégressifs à bifurcation (BAR) ont été au centre de nombreux travaux de recherche ces dernières années. Ces processus, qui sont l'adaptation à un arbre binaire des processus autorégressifs, sont en effet d'intérêt en biologie puisque la structure de l'arbre binaire permet une analogie aisée avec la division cellulaire. L'objectif de cette thèse est l'estimation les paramètres de variantes de ces processus autorégressifs à bifurcation, à savoir les processus BAR à valeurs entières et les processus BAR à coefficients aléatoires. Dans un premier temps, nous nous intéressons aux processus BAR à valeurs entières. Nous établissons, via une approche martingale, la convergence presque sûre des estimateurs des moindres carrés pondérés considérés, ainsi qu'une vitesse de convergence de ces estimateurs, une loi forte quadratique et leur comportement asymptotiquement normal. Dans un second temps, on étudie les processus BAR à coefficients aléatoires. Cette étude permet d'étendre le concept de processus autorégressifs à bifurcation en généralisant le côté aléatoire de l'évolution. Nous établissons les mêmes résultats asymptotiques que pour la première étude. Enfin, nous concluons cette thèse par une autre approche des processus BAR à coefficients aléatoires où l'on ne pondère plus nos estimateurs des moindres carrés en tirant parti du théorème de Rademacher-Menchov. / Bifurcating autoregressive (BAR) processes have been widely investigated this past few years. Those processes, which are an adjustment of autoregressive processes to a binary tree structure, are indeed of interest concerning biology since the binary tree structure allows an easy analogy with cell division. The aim of this thesis is to estimate the parameters of some variations of those BAR processes, namely the integer-valued BAR processes and the random coefficients BAR processes. First, we will have a look to integer-valued BAR processes. We establish, via a martingale approach, the almost sure convergence of the weighted least squares estimators of interest, together with a rate of convergence, a quadratic strong law and their asymptotic normality. Secondly, we study the random coefficients BAR processes. The study allows to extend the principle of bifurcating autoregressive processes by enlarging the randomness of the evolution. We establish the same asymptotic results as for the first study. Finally, we conclude this thesis with an other approach of random coefficient BAR processes where we do not weight our least squares estimators any more by making good use of the Rademacher-Menchov theorem.

Dans cette thèse, nous nous intéressons aux méthodes dites récursives qui permettent une mise à jour des estimations séquentielles de données spatiales ou spatio-temporelles et qui ne nécessitent pas un stockage permanent de toutes les données. Traiter et analyser des flux des données, Data Stream, de façon effective et efficace constitue un défi actif en statistique. En effet, dans beaucoup de domaines d'applications, des décisions doivent être prises à un temps donné à la réception d'une certaine quantité de données et mises à jour une fois de nouvelles données disponibles à une autre date. Nous proposons et étudions ainsi des estimateurs à noyau de la fonction de densité de probabilité et la fonction de régression de flux de données spatiales ou spatio-temporelles. Plus précisément, nous adaptons les estimateurs à noyau classiques de Parzen-Rosenblatt et Nadaraya-Watson. Pour cela, nous combinons la méthodologie sur les estimateurs récursifs de la densité et de la régression et celle d'une distribution de nature spatiale ou spatio-temporelle. Nous donnons des applications et des études numériques des estimateurs proposés. La spécificité des méthodes étudiées réside sur le fait que les estimations prennent en compte la structure de dépendance spatiale des données considérées, ce qui est loin d'être trivial. Cette thèse s'inscrit donc dans le contexte de la statistique spatiale non-paramétrique et ses applications. Elle y apporte trois contributions principales qui reposent sur l'étude des estimateurs non-paramétriques récursifs dans un cadre spatial/spatio-temporel et s'articule autour des l'estimation récursive à noyau de la densité dans un cadre spatial, l'estimation récursive à noyau de la densité dans un cadre spatio-temporel, et l'estimation récursive à noyau de la régression dans un cadre spatial. / In this thesis, we are interested in recursive methods that allow to update sequentially estimates in a context of spatial or spatial-temporal data and that do not need a permanent storage of all data. Process and analyze Data Stream, effectively and effciently is an active challenge in statistics. In fact, in many areas, decisions should be taken at a given time at the reception of a certain amount of data and updated once new data are available at another date. We propose and study kernel estimators of the probability density function and the regression function of spatial or spatial-temporal data-stream. Specifically, we adapt the classical kernel estimators of Parzen-Rosenblatt and Nadaraya-Watson. For this, we combine the methodology of recursive estimators of density and regression and that of a distribution of spatial or spatio-temporal data. We provide applications and numerical studies of the proposed estimators. The specifcity of the methods studied resides in the fact that the estimates take into account the spatial dependence structure of the relevant data, which is far from trivial. This thesis is therefore in the context of non-parametric spatial statistics and its applications. This work makes three major contributions. which are based on the study of non-parametric estimators in a recursive spatial/space-time and revolves around the recursive kernel density estimate in a spatial context, the recursive kernel density estimate in a space-time and recursive kernel regression estimate in space.

L'urne de Polya est l'exemple typique de processus stochastique avec renforcement. La limite presque sûre (p.s.) en temps existe, est aléatoire et non dégénérée. L'urne de Friedman est une généralisation naturelle dont la limite (proportion asymptotique en temps) n'est plus aléatoire. De nombreux modèles aléatoires sont fondés sur des processus de renforcement comme pour la conception d'essais cliniques au design adaptatif, en économie, ou pour des algorithmes stochastiques à des fins d'optimisation ou d'estimation non paramétrique. Dans ce mémoire, inspirés par de nombreux articles récents, nous introduisons une nouvelle famille de systèmes (finis) de processus de renforcement où l'interaction se traduit par un phénomène de renforcement collectif additif, de type champ moyen. Les deux taux de renforcement (l'un spécifique à chaque composante, l'autre collectif et commun à toutes les composantes) sont possiblement différents. Nous prouvons deux types de résultats mathématiques. Différents régimes de paramètres doivent être considérés : type de la règle (brièvement, Polya/Friedman), taux du renforcement. Nous prouvons l'existence d'une limite p.s. coommune à toutes les composantes du système (synchronisation). La nature de la limite (aléatoire/déterministe) est étudiée en fonction du régime de paramètres. Nous étudions également les fluctuations en prouvant des théorèmes centraux de la limite. Les changements d'échelle varient en fonction du régime considéré. Différentes vitesses de convergence sont ainsi établies. / The Polya urn is the paradigmatic example of a reinforced stochastic process. It leads to a random (non degenerated) almost sure (a.s.) time-limit.The Friedman urn is a natural generalization whose a.s. time-limit is not random anymore. Many stochastic models for applications are based on reinforced processes, like urns with their use in adaptive design for clinical trials or economy, stochastic algorithms with their use in non parametric estimation or optimisation. In this work, in the stream of previous recent works, we introduce a new family of (finite) systems of reinforced stochastic processes, interacting through an additional collective reinforcement of mean field type. The two reinforcement rules strengths (one componentwise, one collective) are tuned through (possibly) different rates. In the case the reinforcement rates are like 1/n, these reinforcements are of Polya or Friedman type as in urn contexts and may thus lead to limits which may be random or not. We state two kind of mathematical results. Different parameter regimes needs to be considered: type of reinforcement rule (Polya/Friedman), strength of the reinforcement. We study the time-asymptotics and prove that a.s. convergence always holds. Moreover all the components share the same time-limit (synchronization). The nature of the limit (random/deterministic) according to the parameters' regime is considered. We then study fluctuations by proving central limit theorems. Scaling coefficients vary according to the regime considered. This gives insights into the different rates of convergence.

Neste trabalho estudamos os type-dependent stochastic spin models propostos por Fernández et al., os que chamaremos de modelos de spins estocástico dependentes de tipo, e que foram usados para modelar redes de sinalização biológica. A modelagem original descreve a evolução macroscópica de um modelo de spin-flip de tamanho finito com k tipos de spins, possuindo um número arbitrário de estados internos, que interagem através de uma dinâmica estocástica não reversível. No limite termodinânico foi provado que, em um intervalo de tempo finito as trajetórias convergem quase certamente para uma trajetória determinística, dada por uma equação diferencial de primeira ordem. Os comportamentos destes sistemas dinâmicos podem incluir bifurcações, relacionadas às transições de fase do modelo. O nosso objetivo principal foi de estender os modelos de spins com dinâmica de Glauber utiliza- dos pelos autores, permitindo trocas múltiplas dos spins. No contexto biológico tentamos incluir situações nas quais moléculas de tipos diferentes trocam simultaneamente os seus estados internos. Utilizando diversas técnicas, como as de grandes desvíos e acoplamento, tem sido possível demonstrar a convergência para o sistema dinâmico associado. / We study type-dependent stochastic spin models proposed by Fernández et al., which were used to model biological signaling networks. The original modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. In the thermodynamic limit it was proved that, within arbitrary finite time-intervals, the path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation. The behavior of the associated dynamical system may include bifurcations, associated to phase transitions in the statistical mechanical setting. Our aim is to extend the spin model with Glauber dynamics, to allow multiple spin-flips. In the biological context we included situations in which molecules of different types simultaneously change their internal states. Using several methods, such as large deviations and coupling, we prove the convergence theorem.

Neste trabalho estudamos os type-dependent stochastic spin models propostos por Fernández et al., os que chamaremos de modelos de spins estocástico dependentes de tipo, e que foram usados para modelar redes de sinalização biológica. A modelagem original descreve a evolução macroscópica de um modelo de spin-flip de tamanho finito com k tipos de spins, possuindo um número arbitrário de estados internos, que interagem através de uma dinâmica estocástica não reversível. No limite termodinânico foi provado que, em um intervalo de tempo finito as trajetórias convergem quase certamente para uma trajetória determinística, dada por uma equação diferencial de primeira ordem. Os comportamentos destes sistemas dinâmicos podem incluir bifurcações, relacionadas às transições de fase do modelo. O nosso objetivo principal foi de estender os modelos de spins com dinâmica de Glauber utiliza- dos pelos autores, permitindo trocas múltiplas dos spins. No contexto biológico tentamos incluir situações nas quais moléculas de tipos diferentes trocam simultaneamente os seus estados internos. Utilizando diversas técnicas, como as de grandes desvíos e acoplamento, tem sido possível demonstrar a convergência para o sistema dinâmico associado. / We study type-dependent stochastic spin models proposed by Fernández et al., which were used to model biological signaling networks. The original modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. In the thermodynamic limit it was proved that, within arbitrary finite time-intervals, the path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation. The behavior of the associated dynamical system may include bifurcations, associated to phase transitions in the statistical mechanical setting. Our aim is to extend the spin model with Glauber dynamics, to allow multiple spin-flips. In the biological context we included situations in which molecules of different types simultaneously change their internal states. Using several methods, such as large deviations and coupling, we prove the convergence theorem.

Nous nous intéressons dans cette thèse à certains modèles mathématiques permettant une description de systèmes ouverts classiques et quantiques. Dans l'étude de ces systèmes en interaction avec un environnement, nous montrons que la dynamique induite par l'environnement sur le système donne lieu à l'apparition de bruits. Dans une première partie de la thèse, dédiée aux systèmes classiques, le modèle décrit est le schéma d'interactions répétées. Etant à la fois hamiltonien et markovien, ce modèle en temps discret permet d'implémenter facilement la dissipation dans des systèmes physiques. Nous expliquons comment le mettre en place pour des systèmes physiques avant d'en étudier la limite en temps continu. Nous montrons la convergence Lp et presque sûre de l'évolution de certains systèmes vers la solution d'une équation différentielle stochastique, à travers l'étude de la limite de la perturbation d'un schéma d'Euler stochastique. Dans une seconde partie de la thèse sur les systèmes quantiques, nous nous intéressons dans un premier temps aux actions d'environnements quantiques sur des systèmes quantiques aboutissant à des bruits classiques. A cette fin, nous introduisons certains opérateurs unitaires appelés « classiques », que nous caractérisons à l'aide de variables aléatoires dites obtuses. Nous mettons en valeur comment ces variables classiques apparaissent naturellement dans ce cadre quantique à travers des 3-tenseurs possédant des symétries particulières. Nous prouvons notamment que ces 3-tenseurs sont exactement ceux diagonalisables dans une base orthonormée. Dans un second temps, nous étudions la limite en temps continu d'une variante des interactions répétées quantiques dans le cas particulier d'un système biparti, c'est-à-dire composé de deux systèmes isolés sans interaction entre eux. Nous montrons qu'à la limite du temps continu, une interaction entre ces sous-systèmes apparaît explicitement sous forme d'un hamiltonien d'interaction; cette interaction résulte de l'action de l'environnement et de l'intrication qu'il crée / This dissertation is dedicated to some mathematical models describing classical and quantum open systems. In the study of these systems interacting with an environment, we particularly show that the dynamics induced by the environment leads to the appearance of noises. In a first part of this thesis, devoted to classical open systems, the repeated interaction scheme is developed. This discrete-time model, being Hamiltonian and Markovian at the same time, has the advantage to easily implement the dissipation in physical systems. We explain how to set this scheme up in some physical examples. Then, we investigate the continuous-time limit of these repeated interactions. We show the Lp and almost sure convergences of the evolution of the system to the solution of a stochastic differential equation, by studying the limit of a perturbed Stochastic Euler Scheme. In a second part of this dissertation on quantum systems, we characterize in a first work classical actions of a quantum environment on a quantum system. In this study, we introduce some “classical” unitary operators representing these actions and we highlight a strong link between them and some random variables, called obtuse random variables. We explain how these random variables are naturally connected to some 3-tensors having some particular symmetries. We particularly show that these 3 tensors are exactly the ones that are diagonalizable in some orthonormal basis. In a second work of this part, we study the continuous-time limit of a variant of the repeated interaction scheme in a case of a bipartite system, that is, a system made of two isolated systems not interaction together. We prove that an explicit Hamiltonian interaction between them appears at the limit. This interaction is due to the action of the environment and the entanglement between the two systems that it creates