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111	Théorèmes asymptotiques pour les équations de Boltzmann et de Landau / Asymptotic theorems for Boltzmann and Landau equations Carrapatoso, Kléber 09 December 2013 (has links) Nous nous intéressons dans cette thèse à la théorie cinétique et aux systèmes de particules dans le cadre des équations de Boltzmann et Landau. Premièrement, nous étudions la dérivation des équations cinétiques comme des limites de champ moyen des systèmes de particules, en utilisant le concept de propagation du chaos. Plus précisément, nous étudions les probabilités chaotiques sur l'espace de phase de ces systèmes de particules : la sphère de Boltzmann, qui correspond à l'espace de phase d'un système de particules qui évolue conservant le moment et l'énergie ; et la sphère de Kac, correspondant à un système de particules qui conserve seulement l'énergie. Ensuite, nous nous intéressons à la propagation du chaos, avec des estimations quantitatives et uniforme en temps, pour les équations de Boltzmann et Landau. Deuxièmement, nous étudions le comportement asymptotique en temps grand des solutions de l'équation de Landau. / This thesis is concerned with kinetic theory and many-particle systems in the setting of Boltzmann and Landau equations. Firstly, we study the derivation of kinetic equation as mean field limits of many-particle systems, using the concept of propagation of chaos. More precisely, we study chaotic probabilities on the phase space of such particle systems : the Boltzmann's sphere, which corresponds to the phase space of a many-particle system undergoing a dynamics that conserves momentum and energy ; and the Kac's sphere, which corresponds to the energy conservation only. Then we are concerned with the propagation of chaos, with quantitative and uniform in time estimates, for Boltzmann and Landau equations. Secondly, we study the long-time behaviour of solutions to the Landau equation. Théorie cinétique Systèmes de particules Équation de Landau Équation de Boltzmann Processus à sauts Chaos Chaos entropique Fisher chaos Propagation du chaos Entropie Théorème central limite Retour à l'équilibre Convergence exponentielle Trou spectral Hypodissipativité Molécules maxwelliennes Potentiels durs Collisions rasantes Kinetic theory Many-particle system Landau equation Boltzmann equation Jump process Mean field limit Chaos Entropic chaos Fisher chaos Propagation of chaos Entropy Central limit theorem Relaxation to equilibrium Exponential convergence Spectral gap Hypodissipativity Maxwellian molecules Hard potentials Grazing collisions 510
112	Introduction to Probability Theory Chen, Yong-Yuan 25 May 2010 (has links) In this paper, we first present the basic principles of set theory and combinatorial analysis which are the most useful tools in computing probabilities. Then, we show some important properties derived from axioms of probability. Conditional probabilities come into play not only when some partial information is available, but also as a tool to compute probabilities more easily, even when partial information is unavailable. Then, the concept of random variable and its some related properties are introduced. For univariate random variables, we introduce the basic properties of some common discrete and continuous distributions. The important properties of jointly distributed random variables are also considered. Some inequalities, the law of large numbers and the central limit theorem are discussed. Finally, we introduce additional topics the Poisson process. Chebyshev¡¦s inequality central limit theorem Poisson process Markov¡¦s inequality multivariate normal distribution multinomial distribution F distribution t distribution chi-square distribution lognormal distribution Beta distribution Cauchy distribution Weibull distribution gamma distribution exponential distribution normal distribution discrete uniform distribution uniform distribution hypergeometric distribution geometric distribution negative binomial distribution Poisson distribution binomial distribution Bernoulli distribution Bayes theorem theorem of total probability axioms of probability multinomial theorem inclusion-exclusion principle set
113	Statistical Inference Chou, Pei-Hsin 26 June 2008 (has links) In this paper, we will investigate the important properties of three major parts of statistical inference: point estimation, interval estimation and hypothesis testing. For point estimation, we consider the two methods of finding estimators: moment estimators and maximum likelihood estimators, and three methods of evaluating estimators: mean squared error, best unbiased estimators and sufficiency and unbiasedness. For interval estimation, we consider the the general confidence interval, confidence interval in one sample, confidence interval in two samples, sample sizes and finite population correction factors. In hypothesis testing, we consider the theory of testing of hypotheses, testing in one sample, testing in two samples, and the three methods of finding tests: uniformly most powerful test, likelihood ratio test and goodness of fit test. Many examples are used to illustrate their applications. finite population correction factor statistical hypothesis type I error composite hypothesis power Central limit theorem Basu theorem critical value goodness of fit test likelihood ratio test Lehmann-Scheffe theorem complete statistic ancillary statistic P-value type II error Rao-Blackwell theorem simple hypothesis one-sided test two-sided test significance level sufficient statistic alternative hypothesis testing hypothesis null hypothesis rejection region confidence interval confidence level minimum variance unbiased estimator interval estimation uniformly most powerful test Cramer-Rao inequality likelihood function mean square error moment estimator error of estimation point estimation maximum likelihood estimator bias consistent estimator minimal sufficient statistic factorization theorem unbiased estimator statistic most powerful test test statistics Neyman-Pearson lemma decision rule

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