Consider a simple model of an age-structured population with two age-classes and stochastically varying survival rate of young. Let $m_{1,y},m_{2,t}$ be birth rates per capital and $P_{1,t}$ be a survival rate. egin{eqnarray} left( egin{array}{clr} N_{1,t+1}N_{2,t+1} end{array} ight) = left( egin{array}{clr} m_{1,t+1} & m_{2,t+1} P_{1,t+1} & 0 end{array} ight) left( egin{array}{clr} N_{1,t}N_{2,t} end{array} ight) end{eqnarray} we want to study the large term behavior of $(N_{1,t},N_{2,t})$ the age-structured population through the theory of random matrix product.

In this expository paper we study the stationary measures of a stochastic process called nearest neighbor random walk on Z, and further we describe conditions for these measures to have the stronger property of reversibility. We consider both the cases of symmetric and non-symmetric random walk.

In 2 ¡Ñ2 case,we discuss the uniqueness of the u-invariant measure on projective space.Under the condition that |detM|=1 for any M in Gu and Gu is not compact,we have the followings: (1) For any x in P(R^2),if #{M¡Dx|M belongs Gu}>2, then the u-invariant measure is unique. (2) For some x in P(R^2),there exists x1,x2 such that {M¡Dx|M belongs Gu} is contained in {x1,x2},if x1 and x2 are both fixed,then the u-invariant measure v is not unique;otherwise,if u has mass only on x1 and x2,then the u-invariant measure is unique.

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Neste trabalho estudamos medidas invariantes para aplicações unimodais. Estamos especialmente interessados em detectar as situações que levam uma aplicação unimodal a não possuir uma medida piac, ou seja, uma medida de probabilidade invariante e absolutamente contínua em relação à medida de Lebesgue. Mostramos que a ordem do ponto crítico e a sua capacidade de recorrência são os fatores mais relevantes nesta questão. Os valores das derivadas da aplicação nos pontos periódicos tem uma infuência menor, mas suficiente para garantir que numa mesma classe de conjuga ção topológica podem existir duas aplicações unimodais com ponto crítico de mesma ordem, sendo que uma delas possui medida piac e a outra não possui. A capacidade de recorrência do ponto crítico, talvez o principal fator nesta questão, depende de aspectos combinatórios bem sofisticados. As ferramentas principais para analisar estes aspectos envolvem os conceitos de tempos de corte e de aplicações kneading. A existência ou não de medidas piac é uma propriedade de natureza métrica, e por isto, é necessário que tenhamos controle de como os iterados da aplicação unimodal distorcem a medida de Lebesgue. Então precisamos usar ferramentas de controle de distorção que incluem principalmente os Princípios de Koebe. Um ponto culminante deste trabalho diz respeito a relação entre existência de mediada piac e existência de atratores selvagens, isto é: atratores métricos que não são atratores topológicos e vice versa. Usamos aqui um argumento probabilístico de rara beleza. / In this work we study invariant measures for unimodal maps. We are especially interested in detecting situations that cause a unimodal map not to have a piac measure, i.e., a measure that is Probability Invariant and Absolutely Continuous with respect to Lebesgue measure. We show that the order of the critical point and its capacity for recurrence are the most relevant factors in this matter. The values of the derivatives of the map at periodic points have a small inuence, but enough to ensure that within a single class of topological conjugacy, there can be two unimodal maps with critical points of the same order, one of which has a piac measure and the other does not. The recurrence capacity of the critical point depends on very sophisticated combinatorial aspects and is probably the main factor in this issue. The main tools to analyze these aspects involve the concepts of cutting times and kneading maps. The existence of piac measures is a property of metric nature, and for this reason we need to have control of how iterations of the unimodal map distort the Lebesgue measure. We therefore need to use distortion control tools, including especially the Principles of Koebe. A culmination of this work concerns the relationship between existence of piac measures and the existence of wild attractors, i.e., metric attractors that not are topological attractors. Here we use a probabilistic argument of rare beauty.

Neste trabalho estudamos medidas invariantes para aplicações unimodais. Estamos especialmente interessados em detectar as situações que levam uma aplicação unimodal a não possuir uma medida piac, ou seja, uma medida de probabilidade invariante e absolutamente contínua em relação à medida de Lebesgue. Mostramos que a ordem do ponto crítico e a sua capacidade de recorrência são os fatores mais relevantes nesta questão. Os valores das derivadas da aplicação nos pontos periódicos tem uma infuência menor, mas suficiente para garantir que numa mesma classe de conjuga ção topológica podem existir duas aplicações unimodais com ponto crítico de mesma ordem, sendo que uma delas possui medida piac e a outra não possui. A capacidade de recorrência do ponto crítico, talvez o principal fator nesta questão, depende de aspectos combinatórios bem sofisticados. As ferramentas principais para analisar estes aspectos envolvem os conceitos de tempos de corte e de aplicações kneading. A existência ou não de medidas piac é uma propriedade de natureza métrica, e por isto, é necessário que tenhamos controle de como os iterados da aplicação unimodal distorcem a medida de Lebesgue. Então precisamos usar ferramentas de controle de distorção que incluem principalmente os Princípios de Koebe. Um ponto culminante deste trabalho diz respeito a relação entre existência de mediada piac e existência de atratores selvagens, isto é: atratores métricos que não são atratores topológicos e vice versa. Usamos aqui um argumento probabilístico de rara beleza. / In this work we study invariant measures for unimodal maps. We are especially interested in detecting situations that cause a unimodal map not to have a piac measure, i.e., a measure that is Probability Invariant and Absolutely Continuous with respect to Lebesgue measure. We show that the order of the critical point and its capacity for recurrence are the most relevant factors in this matter. The values of the derivatives of the map at periodic points have a small inuence, but enough to ensure that within a single class of topological conjugacy, there can be two unimodal maps with critical points of the same order, one of which has a piac measure and the other does not. The recurrence capacity of the critical point depends on very sophisticated combinatorial aspects and is probably the main factor in this issue. The main tools to analyze these aspects involve the concepts of cutting times and kneading maps. The existence of piac measures is a property of metric nature, and for this reason we need to have control of how iterations of the unimodal map distort the Lebesgue measure. We therefore need to use distortion control tools, including especially the Principles of Koebe. A culmination of this work concerns the relationship between existence of piac measures and the existence of wild attractors, i.e., metric attractors that not are topological attractors. Here we use a probabilistic argument of rare beauty.

The main topic of this Thesis is a new simplified proof of the Sunyach theorem that provides suffici- ent conditions for existence and uniqueness of an invariant measure for a Markov kernel on a complete separable metric space equipped with its Borel σ-algebra. Weak convergence of measures following from Sunyach's theorem is strengthened to convergence in the total variation norm provided that the Markov kernel is strong Feller. Furthermore, sufficient conditions for geometric ergodicity are stated. Another topic treated is the strong Feller property: its characterization by absolute measurability and uniform integrability and derivation of some other sufficient conditions.

This thesis consists of two parts. We start with some background theory that will be used throughout the thesis. Then, in the first part, we investigate the existence and uniqueness of the solution of the stochastic partial differential equation with two reflecting walls. Then we establish the existence and uniqueness of invariant measure of this equation under some reasonable conditions. In the second part, we study the splitting-up method for approximating the solu- tions of stochastic Stokes equations using resolvent method.

On choisit un groupe algébrique G, un sous-groupe algébrique H de G ; on choisit une mesure de probabilité borélienne μ sur G. On considère alors la chaîne de Markov sur l’espace homogène X = G/H de probabilité de transition Px = μ * δx pour x ε X. Dans cette thèse, on étudie les propriétés de récurrence de ces marches aléatoires.On s’intéresse à deux types de récurrence : la récurrence presque-sûre (toute trajectoire revient presque-sûrement infiniment souvent dans un compact) et la récurrence en loi (il existe une mesure de probabilité μ stationnaire sur X .On s’intéresse également aux éventuelles propriétés de transience presque-sûre (toute trajectoire quitte presque-sûrement définitivement tout compact).On construira d’abord un exemple où on n’a ni récurrence presque-sûre en tout point, ni transience presque-sûre en tout point. On montrera ensuite un critère de récurrence presque-sûre dans le cas où G est un groupe de Lie semi-simple ; on a en fait dans ce cas une dichotomie : soit tous les points sont récurrents,soit tous les points sont transients.Dans le cas où G est le groupe affine GL(d,ℝ) α ℝd,on donnera un critère de récurrence en loi sur les Grassmanniennes affines, et, dans un dernier chapitre, on donnera quelques résultats partiels d'un projet en cours,permettant de donner des résultats pour le groupe SO(p, p+1) α ℝ2p+1. / Choose an algebraic group G, and an algebraic subgroup H. Choose a Borel probability measure μ on G. Consider the Markov chain on the G-space X = G/H with transition probability Px = μ * δx for x ε X.The point of this dissertation is the study of the recurrence properties of such a random walk.We consider two types of recurrence : almost-certain recurrence (i.e. almost-every trajectory enters some compact set infinitely often) and the associated almost-certain transience (where almost-every trajectory eventually leaves every compact set) and recurrence in law (i.e. there exists a μ stationary probability measure on X).First, we show that, in general, there is no dichotomy between almost-certain recurrence and transience by constructing an example with both almost-certainly recurrent and almost-certainly transient points.We then prove a criterion for almost-certain recurrence when G is a semi-simple Lie group and X is a G-space. In fact, in this case, we have a dichotomy where either every point of X is almost-certainly recurrent, or every point of X is almost certainly transient.When G is the affine group GL(d,ℝ) α ℝd, we give a criterion for recurrence in law on the affine Grassmannians.In the final chapter, we give some partial results from an ongoing project,which give a criterion for recurrence in law the group SO(p,p+1)α ℝ2p+1.

This thesis develops a methodology for the construction of generalized families of probability distributions in the quantile statistical universe, that is, distributions specified in terms of their quantile functions. The main benefit of the proposed methodology is that it generates quantile-based distributions with skewness-invariant measures of kurtosis. The skewness and kurtosis can therefore be identified and analyzed separately. The key contribution of this thesis is the development of a new type of the generalized lambda distribution (GLD), using the quantile function of the generalized Pareto distribution as the basic building block (in the literature each different type of the GLD is incorrectly referred to as a parameterization of the GLD – in this thesis the term type is used). The parameters of this new type can, contrary to existing types, easily be estimated with method of L-moments estimation, since closed-form expressions are available for the estimators as well as for their asymptotic standard errors. The parameter space and the shape properties of the new type are discussed in detail, including its characterization through L-moments. A simple estimation algorithm is presented and utilization of the new type in terms of data fitting and approximation of probability distributions is illustrated. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Statistics / unrestricted