Cette thèse s'intéresse à deux modèles de processus auto intéagissant étroitement reliés: le processus de sauts renforcé par sites (VRJP) et la marche aléatoire renforcée par arêtes (ERRW). Nous étudions aussi les liens entre ces processus et un opérateur de Schrödinger aléatoire. Dans le chapitre 3, nous montrons que le VRJP est le seul processus satisfaisant la propriété d'échangeabilité partielle et tel que la probabilité de transition ne dépende que du temps local des voisins, sous quelques conditions techniques. Le chapitre 4 donne la transition de phase entre vitesse positive et vitesse nulle pour un VRJP transitoire sur un arbre de Galton Watson, utilisant le fait que sur un arbre, le VRJP est une marche aléatoire en milieu aléatoire. Dans le chapitre 5, une nouvelle famille exponentielle de loi est introduite et ses liens avec le VRJP sont étudiés. En particulier, nous donnons une preuve de la formule de Coppersmith et Diaconis, n'utilisant que des calculs élémentaires. Finalement, dans le chapitre 6 nous étudions la représentation du VRJP comme mélange de processus de Markov sur les graphes infinis. Nous représentons le VRJP à l'aide de la fonction de Green et d'une fonction propre généralisée d'un opérateur de Schrödinger aléatoire associé au VRJP. En conséquence, nous obtenons un principe d'invariance pour le VRJP quand le renforcement est suffisamment faible, ainsi que la récurrence du ERRW sur ℤ2 pour toute valeurs initiales des paramètres / This thesis is dedicated to the study of two closely related self-interacting processes: the vertex reinforced jump process (VRJP) and the edge reinforced random walk (ERRW). We also study the relations between these processes and a random Schrödinger operator. In Chapter 3, we prove that the VRJP is the only partially exchangeable process whose transition probability depends only on neighbor local times, under some technical conditions. Chapter 4 gives the phase transition between positive speed and null speed of a transient VRJP on a Galton Watson tree, using a representation of random walk in independent random environment. In Chapter 5, we introduce a new exponential family of probability distributions generalizing the Inverse Gaussian distribution, and we show some of its relations to the VRJP. In particular, we give an elementary proof of the formula of Coppersmith and Diaconis. Finally, we show in Chapter 6 that the VRJP on infinite graph is a mixture of Markov jump processes, by constructing the random environment using the Green function and a generalized eigenfunction related to a random Schrödinger operator associated with the VRJP. As a consequence, we obtain a central limit theorem when the reinforcement is weak enough, and also the recurrence of ERRW on ℤ2 for any initial constant weights

Motivated by dimension reduction in regression analysis and signal detection, we investigate order determination for large dimensional matrices with spiked structures in which the dimensions of the matrices are proportional to the sample sizes. Because the asymptotic behaviors of the estimated eigenvalues differ completely from those in fixed dimension scenarios, we then discuss the largest possible order, say q, we can identify and introduce criteria for different settings of q. When q is assumed to be fixed, we propose a "valley-cliff" criterion with two versions - one based on the original differences of eigenvalues and the other based on the transformed differences - to reduce the effect of ridge selection in the criterion. This generic method is very easy to implement and computationally inexpensive, and it can be applied to various matrices. As examples, we focus on spiked population models, spiked Fisher matrices and factor models with auto-covariance matrices. For the case of divergent q, we propose a scale-adjusted truncated double ridge ratio (STDRR) criterion, where a scale adjustment is implemented to deal with the bias in scale parameter for large q. Again, examples include spiked population models, spiked Fisher matrices. Numerical studies are conducted to examine the finite sample performances of the method and to compare it with existing methods. As for theoretical contributions, we investigate the limiting properties, including convergence in probability and central limit theorems, for spiked eigenvalues of spiked Fisher matrices with divergent q. Keywords: Auto-covariance matrix, factor model, finite-rank perturbation, Fisher matrix, principal component analysis (PCA), phase transition, random matrix theory (RMT), ridge ratio, spiked population model.

Random fiber lasers, a new type of fiber laser that uses disordered medium to provide distributed feedback, have drawn considerable interest in the photonics community over the past ten years. Stimulated Brillouin scattering (SBS), with a typical narrow spectral width of ~100 MHz, provides an important gain mechanism for random fiber lasers. Brillouin random fiber laser (BRFL) has shown excellent advantages in generating highly coherent photons and in ultrasound sensing. However, the accompanied large intensity noise in BRFLs hinders its further performance improvement and practical applications. In order to design a low noise BRFL, it is important to explore the fundamental physics behind BRFL and study its output characteristics. This thesis focuses on the study of random lasing mechanism in BRFL, which lays the foundation for the demonstration of a low noise BRFL. The main research results and contributions are as follows: (1) In order to understand the dynamic noise properties of BRFLs, the properties of the acoustic wave generated by BRFL, including its intrinsic spectral width, intensity dynamics, distributed spectrum and distributed intensity statistics are characterized for the first time. The characterization method is based on the SBS enhanced polarization decoupled four wave mixing process, where the pump wave, Stokes wave, probe wave and reflected probe wave are coupled through the fiber density variation induced by the acoustic wave. It is demonstrated that the intrinsic spectral width of the acoustic wave in the Brillouin gain fiber depends on the spectral convolution of pump light and Stokes light. Stochastic behaviour is introduced to the intensity dynamics of the acoustic wave when the linewidth of the pump light (or the Stokes light) is larger than several MHz. The distributed spectra of the dynamic grating are determined by the birefringence of the Brillouin gain fiber, which have maximum change on the order of 10-7 to 10-6 when the BRFL is on operation. Different proportion of optical rogue waves are detected at high gain position and low gain position near the lasing threshold, proving the nonlinear amplification of the SBS process. (2) In order to study the mode selection mechanism of the distributed random feedback and explore new physics phenomenon in BRFLs, the conventional Rayleigh scattering fiber in BRFL is replaced by the artificially controlled random scattering medium. First, weak FBG array with random spacing offers distributed feedback with varied length, which demonstrate the longitudinal mode filter function of the distributed random feedback. Single longitudinal mode operation of BRFL is realized by using appropriate length of the FBG array. Then, scattering from random fiber grating (RFG) with varied grating period is used to provide feedback for BRFL. The enhanced backscattering strength from RFG improves the slope efficiency of BRFL to 29.3% and reduces the lasing threshold to 10.2 mW. By calculating the correlation of the intensity fluctuation spectra from trace to trace, the correlation of two traces is found to be dependent on the specific two chosen traces, demonstrating the replica symmetry breaking phenomenon in photonics. (3) RFG with relatively large refractive index modulation shows potentials in improving the performance of the BRFL. In order to investigate the working mechanism of the RFG, optical frequency domain reflectometry (OFDR) with spatial resolution of 8 μm is employed to characterize the property of RFG. The backscattering strength and spectral response of RFG is highly related to the degree of randomness of RFG. Theoretically, entropy is introduced to build a quantitative relationship between the degree of randomness and backscattering strength of the RFG based on the transfer matrix method. A linear relationship between the average reflectivity of the RFG in dB scale and sub-grating’s entropy is found. Further, based on a polarization maintaining RFG, a low noise BRFL is proposed and demonstrated. Compared to Rayleigh scattering, the polarization maintaining RFG can tolerate environmental perturbation, leading to a 20 dB intensity noise suppression of the BRFL in the low frequency domain from 10 Hz to 1 kHz. (4) The dynamic properties of the slowly varying frequency drift of a dual-wavelength BRFL in polarization maintaining fiber are characterized. Two principal lasing peaks in each polarization are enabled by the combined distributed Rayleigh scattering and the Brillouin gain provided by the polarization maintaining fiber with large birefringence. Polarization dependent and polarization independent spectral variations are studied in the dual-wavelength BRFL due to the environmental perturbation and gain competition. The probability distribution of the lasing frequency exhibits a dip near the mean frequency that is caused by the spectral hole burning. By calculating the matrix of the Pearson correlation coefficient, the internal correlations between different part of random fiber laser spectra are found, which enhances the understanding of the fundamental physics of random lasing process.

Over the years, people have found Quantum Mechanics to be extremely useful in explaining various physical phenomena from a microscopic point of view. Anderson localization, named after physicist P. W. Anderson, states that disorder in a crystal can cause non-spreading of wave packets, which is one possible mechanism (at single electron level) to explain metalinsulator transitions. The theory of quantum computation promises to bring greater computational power over classical computers by making use of some special features of Quantum Mechanics. The first part of this dissertation considers a 3D alloy-type model, where the Hamiltonian is the sum of the finite difference Laplacian corresponding to free motion of an electron and a random potential generated by a sign-indefinite single-site potential. The result shows that localization occurs in the weak disorder regime, i.e., when the coupling parameter λ is very small, for energies E ≤ −Cλ² . The second part of this dissertation considers adiabatic quantum computing (AQC) algorithms for the unstructured search problem to the case when the number of marked items is unknown. In an ideal situation, an explicit quantum algorithm together with a counting subroutine are given that achieve the optimal Grover speedup over classical algorithms, i.e., roughly speaking, reduce O(2n ) to O(2n/2 ), where n is the size of the problem. However, if one considers more realistic settings, the result shows this quantum speedup is achievable only under a very rigid control precision requirement (e.g., exponentially small control error). / Ph. D.

We develop a stochastic control model for an investor's optimal investment and consumption over an uncertain planning horizon when the investor is endowed with a defaultable income stream. The distributions of the random time of default and the random terminal time are prescribed by deterministic hazard rates, and the investor makes investments in a standard financial market with a bond and a stock, modeled by geometric Brownian motion. In addition, the investor purchases insurance against both default and the terminal date, the default insurance serving as a proxy for the investor's disutility for default. We approximate the original continuous-time problem with a sequence of discrete-time Markov chain control problems by applying dynamic programming and the Markov chain approximation. We demonstrate how the problem can be solved numerically through a logarithmic transformation of the investor's wealth variable, even when the utilities are CRRA with large risk aversion parameter. The model and computational approach are applied to a retiree's optimal annuity decision in the presence of default risk, and we demonstrate that default risk can lead a retiree to annuitize significantly smaller proportions of savings, even when a portion of the defaulted annuity can be recovered, than is traditionally considered optimal by the retirement-finance community. Hence, we show that credit risk may play an important role in resolving the annuity puzzle.

With the grant of a big CRM dataset from a large media company, this dissertation examines four different categories of factors that could impact three stages of customer relationship management, namely customer acquisition, retention, and winback of lost customers. Specifically, with the aid of machine learning method of random forests and text mining technique, this study identify among the factors of customer heterogeneity (e.g. in usage of self-care service channels, duration of service, responsiveness to marketing actions), firm's marketing initiatives (e.g. the volume of the marketing communications, the depth of the promotion, the different communication channels they use, and the marketing penetration in different geographical areas), customer self-reported deactivation reasons, as well as the call centers notes in text form, which factors play bigger roles than others during each of the three stages of CRM. Furthermore, the authors also examine how these factors evolve throughout these three stages of CRM in terms of their effects on shaping customers' decision making of whether to convert to paid customer, to churn, or to reactivate their service with the company. The findings help managers better allocate their resources in the processes of acquiring, retaining and winning back customers. / Business Administration/Marketing

No description available.

Se donnant un mécanisme de branchement critique ou sous-critique, on définit une procédure d’élagage de l’arbre aléatoire continu de Lévy associé. Cette procédure d’élagage est définie en plaçant des marques sur l’arbre grâce `a des techniques de serpent de Lévy. On démontre alors que le sous-arbre obtenu après élagage est encore un arbre aléatoire continu de Lévy. Ce résultat est démontré en utilisant une propriété de Markov spéciale et un problème de martingale pour les processus d’exploration. On construit ensuite, par couplage, une autre procédure d’élagage qui définit un processus de fragmentation sur l’arbre. On calcule la famille de mesures de dislocation associée à cette fragmentation. Dans un deuxième travail, on considère une diffusion aléatoire dans un milieu Lévy stable. On montre que le processus des temps locaux renormalisé et recentré au minimum de la vallée standard de hauteur log t, converge en loi vers une fonctionnelle de deux processus de Lévy conditionnés `a rester positifs indépendants. Pour démontrer ce résultat, on montre que la loi de la vallée standard est proche de celle de deux processus de Lévy conditionnés à rester positifs concaténés en 0. On obtient également la loi limite du supremum du temps local renormalisé. / Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using Lévy snake techniques. We then prove that the resulting sub-tree after pruning is still a Lévy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We then construct, by coupling, an another pruning procedure which define a fragmentation process on the tree. We compute the family of dislocation measures associated with this fragmentation. In a second work, we consider a one-dimensional diffusion in a stable Lévy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height log t converges in law to a functional of two independent Lévy processes conditioned to stay positive. To prove this result, we show that the law of the standard valley is close to a two-sided Lévy process conditioned to stay positive. We also obtain the limit law of the supremum of the normalized local time.

We introduce numerical methods for the analysis of random dynamical systems. The subdivision and the continuation algorithm are powerful tools which will be demonstrated for a system from ship dynamics. With our software package we are able to show that the well known safe basin is a moving fractal set. We will also give a numerical approximation of the attracting invariant set (which contains a local attractor) and its evolution.

In this thesis, we study the exact eigenvalue distribution of product of independent rectangular complex Gaussian matrices and also that of product of independent truncated Haar unitary matrices and inverses of truncated Haar unitary matrices. The eigenvalues of these random matrices form determinantal point processes on the complex plane. We also study the limiting expected empirical distribution of appropriately scaled eigenvalues of those matrices as the size of matrices go to infinity. We give the first example of a random matrix whose eigenvalues form a non-rotation invariant determinantal point process on the plane. The second theme of this thesis is infinite products of random matrices. We study the asymptotic behaviour of singular values and absolute values of eigenvalues of product of i .i .d matrices of fixed size, as the number of matrices in the product in-creases to infinity. In the special case of isotropic random matrices, We derive the asymptotic joint probability density of the singular values and also that of the absolute values of eigenvalues of product of right isotropic random matrices and show them to be equal. As a corollary of these results, we show probability that all the eigenvalues of product of certain i .i .d real random matrices of fixed size converges to one, as the number of matrices in the product increases to infinity.