It is believed that in the long time limit, the limiting behavior of the discrete-time quantum random walk will cross from quantum to classical if we take into account of the decoherence. The computer simulation has already shown that for the discrete-time one-dimensional Hadamard quantum random walk with coin decoherence such that the measurement operators on the coin space are defined by A0 = Ic √1 − p, A1 = |R > < R| √p and A2 = |L > < L > < L| √p is diffusive when 0 < p ≤ 1 and it is ballistic when P = 0. In this thesis, we are going to let p to be dynamical depending on the step t, that is, we consider p = 1/tß, ß ≥ 0 and we found that it has sub-ballistic behavior for 0 < ß < 1. Furthermore, we study not only the coin decoherence, but the total decoherence, that means the measurement operators apply on the Hilbert space H = Hp ⊗ Hc instead of the coin space only. We find that the results are both sub-ballistic for the coin and total decoherence when 0 < ß < 1. Moreover, according to the model given in [T. A. Brun, H. A. Carteret, and A. Ambainis, Phys. Rev. A 67, 032304 (2003)], we know that if the walker has chance to hop to the second nearest neighbor lattice in one step, the long-time behavior is also sub-ballistic and it is similar to that the walker can hop to the third nearest neighbor lattice in one step. By the way, we also find that if we combine the classical part of the model given in [Jing Zhao and Peiqing Tong. One-dimensional quantum walks subject to next nearest neighbor hopping decoherence, Nanjing Normal University, preprint (2014)] with different step length, then this decoherence will also cross from quantum to classical. Finally, we define the quantum γ-stable walk and obtain the quantum γ-stable law with decoherence. By this decoherence, we can see that the limiting behavior of the quantum stable walk will also cross from quantum to classical and we shows that it spreads out faster than the classical stable walk. / Mathematics

Les marches aléatoires en milieu aléatoire ont suscité un vif intérêt au cours de ces dernières années, tant en sciences appliquées, comme moyen notamment d'affiner des modèles par une prise en compte des fluctuations de l'environnement, qu'en mathématiques, de par la multiplicité et la richesse des comportements qu'elles présentent. Cette thèse est dédiée à l'étude de divers aspects de la transience des marches aléatoires en milieu aléatoire. Elle est composée de deux parties, la première consacrée au cas des environnements de Dirichlet sur Z^d, la seconde au régime transient sous-diffusif sur Z. La loi de Dirichlet apparaît naturellement du fait de son lien avec les marches renforcées. Certaines de ses spécificités permettent de plus d'obtenir des résultats sensiblement plus précis qu'en général. On démontre ainsi tout d'abord une caractérisation de l'intégrabilité des temps de sortie de parties finies de graphes quelconques, qui permet de raffiner un critère de balisticité dans Z^d. On prouve également que les marches aléatoires en environnement de Dirichlet sont transientes directionnellement, avec probabilité positive, dès que les paramètres ne sont pas symétriques. En dimension 1, la thèse se focalise sur le rôle des vallées profondes de l'environnement, en fournissant une nouvelle preuve du théorème de Kesten-Kozlov-Spitzer dans le cas sous-diffusif basée sur l'étude fine du comportement de la marche. Outre une meilleure compréhension de l'émergence de la loi limite, cette preuve a l'avantage de fournir la valeur explicite de ses paramètres. / Random walks in random environment have raised a great interest in the last few years, both among applied scientists, notably as a way to refine models by taking fluctuations of the surrounding environment into account, and among mathematicians, because of the variety and wealth of behaviours they display. This thesis aims at the study of miscellaneous aspects of the transience of random walks in random environment. A first part is dedicated to Dirichlet environments on Z^d and a second one to the transient subdiffusive regime on Z. Random walks in Dirichlet environment arise naturally as an equivalent model for oriented-edge reinforced reinforced random walks. Its specificities also allow for sensibly sharper results than in the general case. We thus prove a characterization of the integrability of exit times out of finite subsets of arbitrary graphs, which enables us to refine a ballisticity criterion on Z^d. We also prove that these random walks are transient with positive probability as soon as the parameters are non-symmetric. In dimension 1, the thesis focuses on the role of the deep valleys of the environment. We give a new proof of Kesten-Kozlov-Spitzer theorem in the subdiffusive regime based on a fine study of the behaviour of the walk. Together with a better understanding of the origin of the limit law, this proof also provides its explicit parameters.

Seja $\\mathcal X=\\{\\mathcal X_t:\\, t\\geq0,\\, \\mathcal X_0=0\\}$ um passeio aleatório $\\beta$-estável em $\\mathbb Z$ com média zero e com taxas de saltos não-homogêneas $\\{\\tau_i^: i\\in\\mathbb Z\\}$, com $\\beta\\in(1,2]$ e $\\{\\tau_i: i\\in\\mathbb Z\\}$ sendo uma família de variáveis aleatórias independentes com distribuição marginal comum na bacia de atração de uma lei $\\alpha$-estável, com $\\alpha\\in(0,2]$. Nesta tese, obtemos resultados sobre o comportamento do processo $\\mathcal X_t$ para tempos longos, em particular, obtemos seu limite de escala. Quando $\\alpha\\in(0,1)$, o limite de escala é um processo $\\beta$-estável mudado de tempo pela inversa de um outro processo, o qual envolve o tempo local do processo $\\beta$-estável e um independente subordinador $\\alpha$-estável; chamamos o processo resultante de processo quase-estável. Para o caso $\\alpha\\in[1,2]$, o limite de escala é um ordinário processo $\\beta$-estável. Para $\\beta=2$ e $\\alpha\\in(0,1)$, o limite de escala é uma quase-difusão com medida de velocidade aleatória estudada por Fontes, Isopi e Newman (2002). Outros resultados sobre o comportamento de $\\mathcal X$ para tempos longos são envelhecimento e localização. Nós obtemos resultados de envelhecimento integrado e não-integrado para $\\mathcal X$ quando $\\alpha\\in(0,1)$. Relacionado à esses resultados, e possivelmente de interesse independente, consideramos o processo de armadilha definido por $\\{\\tau_{\\mathcal X_t}: t\\geq0\\}$, e obtemos seu limite de escala. Concluímos a tese com resultados sobre localização de $\\mathcal X$. Mostramos que ele pode ser localizado quando $\\alpha\\in(0,1)$, e que não pode ser localizado quando $\\alpha\\in(1,2]$, assim estendendo os resultados de Fontes, Isopi e Newman (1999) para o caso de passeios simples simétricos. / Let $\\mathcal X=\\{\\mathcal X_t:\\, t\\geq0,\\, \\mathcal X_0=0\\}$ be a mean zero $\\beta$-stable random walk on $\\mathbb Z$ with inhomogeneous jump rates $\\{\\tau_i^: i\\in\\mathbb Z\\}$, with $\\beta\\in(1,2]$ and $\\{\\tau_i: i\\in\\mathbb Z\\}$ is a family of independent random variables with common marginal distribution in the basin of attraction of an $\\alpha$-stable law with $\\alpha\\in(0,2]$. In this thesis we derive results about the long time behavior of this process, in particular its scaling limit. When $\\alpha\\in(0,1)$, the scaling limit is a $\\beta$-stable process time-changed by the inverse of another process, involving the local time of the $\\beta$-stable process and an independent $\\alpha$-stable subordinator; the resulting process may be called a quasistable process. For the case $\\alpha\\in[1,2]$, the scaling limit is an ordinary $\\beta$-stable process. For $\\beta=2$ and $\\alpha\\in(0,1)$, the scaling limit is a quasidiffusion with random speed measure studied by Fontes, Isopi and Newman (2002). Other results about the long time behavior of $\\mathcal X$ concern aging and localization. We obtain integrated and non integrated aging results for $\\mathcal X$ when $\\alpha\\in(0,1)$. Related to these results, and possibly of independent interest, we consider the trap process defined as $\\{\\tau_{\\mathcal X_t}: t\\geq0\\}$, and derive its scaling limit. We conclude the thesis with results about localization of $\\mathcal X$. We show that it localizes when $\\alpha\\in(0,1)$, and does not localize when $\\alpha\\in(1,2]$, extending results of Fontes, Isopi and Newman (1999) for the simple symmetric case.

Seja $\\mathcal X=\\{\\mathcal X_t:\\, t\\geq0,\\, \\mathcal X_0=0\\}$ um passeio aleatório $\\beta$-estável em $\\mathbb Z$ com média zero e com taxas de saltos não-homogêneas $\\{\\tau_i^: i\\in\\mathbb Z\\}$, com $\\beta\\in(1,2]$ e $\\{\\tau_i: i\\in\\mathbb Z\\}$ sendo uma família de variáveis aleatórias independentes com distribuição marginal comum na bacia de atração de uma lei $\\alpha$-estável, com $\\alpha\\in(0,2]$. Nesta tese, obtemos resultados sobre o comportamento do processo $\\mathcal X_t$ para tempos longos, em particular, obtemos seu limite de escala. Quando $\\alpha\\in(0,1)$, o limite de escala é um processo $\\beta$-estável mudado de tempo pela inversa de um outro processo, o qual envolve o tempo local do processo $\\beta$-estável e um independente subordinador $\\alpha$-estável; chamamos o processo resultante de processo quase-estável. Para o caso $\\alpha\\in[1,2]$, o limite de escala é um ordinário processo $\\beta$-estável. Para $\\beta=2$ e $\\alpha\\in(0,1)$, o limite de escala é uma quase-difusão com medida de velocidade aleatória estudada por Fontes, Isopi e Newman (2002). Outros resultados sobre o comportamento de $\\mathcal X$ para tempos longos são envelhecimento e localização. Nós obtemos resultados de envelhecimento integrado e não-integrado para $\\mathcal X$ quando $\\alpha\\in(0,1)$. Relacionado à esses resultados, e possivelmente de interesse independente, consideramos o processo de armadilha definido por $\\{\\tau_{\\mathcal X_t}: t\\geq0\\}$, e obtemos seu limite de escala. Concluímos a tese com resultados sobre localização de $\\mathcal X$. Mostramos que ele pode ser localizado quando $\\alpha\\in(0,1)$, e que não pode ser localizado quando $\\alpha\\in(1,2]$, assim estendendo os resultados de Fontes, Isopi e Newman (1999) para o caso de passeios simples simétricos. / Let $\\mathcal X=\\{\\mathcal X_t:\\, t\\geq0,\\, \\mathcal X_0=0\\}$ be a mean zero $\\beta$-stable random walk on $\\mathbb Z$ with inhomogeneous jump rates $\\{\\tau_i^: i\\in\\mathbb Z\\}$, with $\\beta\\in(1,2]$ and $\\{\\tau_i: i\\in\\mathbb Z\\}$ is a family of independent random variables with common marginal distribution in the basin of attraction of an $\\alpha$-stable law with $\\alpha\\in(0,2]$. In this thesis we derive results about the long time behavior of this process, in particular its scaling limit. When $\\alpha\\in(0,1)$, the scaling limit is a $\\beta$-stable process time-changed by the inverse of another process, involving the local time of the $\\beta$-stable process and an independent $\\alpha$-stable subordinator; the resulting process may be called a quasistable process. For the case $\\alpha\\in[1,2]$, the scaling limit is an ordinary $\\beta$-stable process. For $\\beta=2$ and $\\alpha\\in(0,1)$, the scaling limit is a quasidiffusion with random speed measure studied by Fontes, Isopi and Newman (2002). Other results about the long time behavior of $\\mathcal X$ concern aging and localization. We obtain integrated and non integrated aging results for $\\mathcal X$ when $\\alpha\\in(0,1)$. Related to these results, and possibly of independent interest, we consider the trap process defined as $\\{\\tau_{\\mathcal X_t}: t\\geq0\\}$, and derive its scaling limit. We conclude the thesis with results about localization of $\\mathcal X$. We show that it localizes when $\\alpha\\in(0,1)$, and does not localize when $\\alpha\\in(1,2]$, extending results of Fontes, Isopi and Newman (1999) for the simple symmetric case.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.