Master's thesis is focused on implementing modern statistical methods for fitting propability distribution using kernel estimates with regard to the possibilities of their implementation on the PC and the application of specic data sets. Master's thesis is a part of project from MSMT of the Czech Republic no. 1M06047 Center for Quality and Reliability of Production.

Title: Sequential Monte Carlo Methods Author: David Coufal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Viktor Beneš, DrSc. Abstract: The thesis summarizes theoretical foundations of sequential Monte Carlo methods with a focus on the application in the area of particle filters; and basic results from the theory of nonparametric kernel density estimation. The summary creates the basis for investigation of application of kernel meth- ods for approximation of densities of distributions generated by particle filters. The main results of the work are the proof of convergence of kernel estimates to related theoretical densities and the specification of the development of approx- imation error with respect to time evolution of a filter. The work is completed by an experimental part demonstrating the work of presented algorithms by simulations in the MATLABR⃝ computational environment. Keywords: sequential Monte Carlo methods, particle filters, nonparametric kernel estimates

Dans cette thèse nous étudions les transformées de Riesz et les espaces de Hardy associés à un opérateur sur un espace métrique mesuré. Ces deux sujets sont en lien avec des estimations du noyau de la chaleur associé à cet opérateur. Dans les Chapitres 1, 2 et 4, on étudie les transformées quasi de Riesz sur les variétés riemannienne et sur les graphes. Dans le Chapitre 1, on prouve que les quasi transformées de Riesz sont bornées dans Lp pour 1<p<2. Dans le Chapitre 2, on montre que les quasi transformées de Riesz est aussi de type faible (1,1) si la variété satisfait la propriété de doublement du volume et l'estimation sous-gaussienne du noyau de la chaleur. On obtient des résultats analogues sur les graphes dans le Chapitre 4. Dans le Chapitre 3, on développe la théorie des espaces de Hardy sur les espaces métriques mesurés avec des estimations différentes localement et globalement du noyau de la chaleur. On définit les espaces de Hardy par les molécules et par les fonctions quadratiques. On montre tout d'abord que ces deux espaces H1 sont les mêmes. Puis, on compare l'espace Hp défini par par les fonctions quadratiques et Lp. On montre qu'ils sont équivalents. Mais on trouve des exemples tels que l'équivalence entre Lp et Hp défini par les fonctions quadratiques avec l'homogénéité t2 n'est pas vraie. Finalement, comme application, on montre que les quasi transformées de Riesz sont bornées de H1 dans L1 sur les variétés fractales. Dans le Chapitre 5, on prouve des inégalités généralisées de Poincaré et de Sobolev sur les graphes de Vicsek. On montre aussi qu'elles sont optimales. / In this thesis, we mainly study Riesz transforms and Hardy spaces associated to operators. The two subjects are closely related to volume growth and heat kernel estimates. In Chapter 1, 2 and 4, we study Riesz transforms on Riemannian manifold and on graphs. In Chapter 1, we prove that on a complete Riemannian manifold, the quasi Riesz transform is always Lp bounded on for p strictly large than 1 and no less than 2. In Chapter 2, we prove that the quasi Riesz transform is also weak L1 bounded if the manifold satisfies the doubling volume property and the sub-Gaussian heat kernel estimate. Similarly, we show in Chapter 4 the same results on graphs. In Chapter 3, we develop a Hardy space theory on metric measure spaces satisfying the doubling volume property and different local and global heat kernel estimates. Firstly we define Hardy spaces via molecules and via square functions which are adapted to the heat kernel estimates. Then we show that the two H1 spaces via molecules and via square functions are the same. Also, we compare the Hp space defined via square functions with Lp. The corresponding Hp space for p large than 1 defined via square functions is equivalent to the Lebesgue space Lp. However, it is shown that in this situation, the Hp space corresponding to Gaussian estimates does not coincide with Lp any more. Finally, as an application of this Hardy space theory, we proved that quasi Riesz transforms are bounded from H1 to L1 on fractal manifolds. In Chapter 5, we consider Vicsek graphs. We prove generalised Poincaré inequalities and Sobolev inequalities on Vicsek graphs and we show that they are optimal.

Made available in DSpace on 2015-03-03T15:22:33Z (GMT). No. of bitstreams: 1 MariaASS_DISSERT.pdf: 1267496 bytes, checksum: aabe75eb05da3e6d622acfd6e208c192 (MD5) Previous issue date: 2010-02-25 / In this work we studied the consistency for a class of kernel estimates of f f (.) in the Markov chains with general state space E C Rd case. This study is divided into two parts: In the first one f (.) is a stationary density of the chain, and in the second one f (x) v (dx) is the limit distribution of a geometrically ergodic chain / Neste trabalho vamos estudamos a consist?ncia para uma classe de estimadores n?cleo de f (.) em cadeias de Markov com espa?o de estados geral E c Rd. Este estudo ? dividido em duas partes: Na primeira f (.) ? uma densidade estacion?ria de uma cadeia, e no segundo f (x) v (dx) ? a distribui??o limite de uma cadeia geometricamente erg?dica

Made available in DSpace on 2015-03-03T15:28:31Z (GMT). No. of bitstreams: 1 EnaiTC_DISSERT.pdf: 791005 bytes, checksum: 0aee506aa23dd64244f916f2b42bdf10 (MD5) Previous issue date: 2010-05-21 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work, we studied the strong consistency for a class of estimates for a transition density of a Markov chain with general state space ?E ? Rd. The strong ergodicity of the estimates for the density transition is obtained from the strong consistency of the kernel estimates for both the marginal density p(:) of the chain and the joint density q(., .). In this work the Markov chain is supposed to be homogeneous, uniformly ergodic and possessing a stationary density p(.,.) / No presente trabalho, estudamos a consist?ncia forte de uma classe de estimadores do tipo n?cleo para a densidade de transi??o de uma cadeia de Markov com espa?o de estados geral. A consist?ncia forte do estimador da densidade de transi??o, foi obtida a partir da consist?ncia forte dos estimadores do tipo n?cleo para a densidade marginal da cadeia, , e da consist?ncia forte dos estimadores do tipo n?cleo para a densidade conjunta, . Foi considerado que a cadeia ? homog?nea e uniformemente erg?dica com densidade inicial estacion?ria

Made available in DSpace on 2014-12-17T15:26:39Z (GMT). No. of bitstreams: 1 MarianaBS_DISSERT.pdf: 5316617 bytes, checksum: 4fb0344851aa8f373aa2dab90bb6d3c5 (MD5) Previous issue date: 2012-05-31 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work, the paper of Campos and Dorea [3] was detailed. In that article a Kernel Estimator was applied to a sequence of random variables with general state space, which were independent and identicaly distributed. In chapter 2, the estimator?s properties such as asymptotic unbiasedness, consistency in quadratic mean, strong consistency and asymptotic normality were verified. In chapter 3, using R software, numerical experiments were developed in order to give a visual idea of the estimate process / Neste trabalho estudamos um dos m?todos n?o-param?trico: os Estimadores do Tipo N?cleo associado a uma sequ?ncia de vari?veis aleat?rias independentes e identicamente distribu?das com espa?o de estados geral, mais precisamente o trabalho de Campos e Dorea [3]. No Cap?tulo 2 verificamos as boas qualidades dessa classe de estimadores como n?o v?cio assint?tico, converg?ncia em m?dia quadr?tica, consist?ncia forte e normalidade assint?tica. No Cap?tulo 3 com o auxilio do software R temos uma id?ia visual do que ocorre no processo de estima??o

In this thesis, we present a new existence result for Lévy-type processes. Lévy-type processes behave locally like a Lévy process, but the Lévy triplet may depend on the current position of the process. They can be characterized by their so-called symbol; this is the analogue of the characteristic exponent in the Lévy case. Using a parametrix construction, we prove the existence of Lévy-type processes with a given symbol under weak regularity assumptions on the regularity of the symbol. Applications range from existence results for stable-like processes and mixed processes to uniqueness results for Lévy-driven stochastic differential equations. Moreover, we discuss sufficient conditions for the existence of moments of Lévy-type processes and derive estimates for fractional moments.

In this thesis, we present a new existence result for Lévy-type processes. Lévy-type processes behave locally like a Lévy process, but the Lévy triplet may depend on the current position of the process. They can be characterized by their so-called symbol; this is the analogue of the characteristic exponent in the Lévy case. Using a parametrix construction, we prove the existence of Lévy-type processes with a given symbol under weak regularity assumptions on the regularity of the symbol. Applications range from existence results for stable-like processes and mixed processes to uniqueness results for Lévy-driven stochastic differential equations. Moreover, we discuss sufficient conditions for the existence of moments of Lévy-type processes and derive estimates for fractional moments.