概率論教程

內(nèi)容概要

　　《概率論教程》是一部講述現(xiàn)代概率論及其測度論應(yīng)用基礎(chǔ)的教程，其目標(biāo)讀者是該領(lǐng)域的研究生和相關(guān)的科研人員。內(nèi)容廣泛，有許多初級教程不能涉及到得的。理論敘述嚴(yán)謹(jǐn)，獨立性強(qiáng)。有關(guān)測度的部分和概率的章節(jié)相互交織，將概率的抽象性完全呈現(xiàn)出來。此外，還有大量的圖片、計算模擬、重要數(shù)學(xué)家的個人傳記和大量的例子。這使得表現(xiàn)形式更加活躍。

作者簡介

作者:(德)凱蘭克

書籍目錄

preface 1 basic measure theory 1.1 classes of sets 1.2 set functions 1.3 the measure extension theorem 1.4 measurable maps 1.5 random variables 2 independence 2.1 independence of events 2.2 independent random variables 2.3 kolmogorov's 0-1 law 2.4 example: percolation 3 generating functions 3.1 definition and examples 3.2 poisson approximation 3.3 branching processes 4 the integral 4.1 construction and simple properties 4.2 monotone convergence and fatou's lemma .4.3 lebesgue integral versus riemann integral 5 moments and laws of large numbers 5.1 moments 5.2 weak law of large numbers 5.3 strong law of large numbers 5.4 speed of convergence in the strong lln 5.5 the poisson process 6 convergence theorems 6.1 almost sure and measure convergence 6.2 uniform integrability 6.3 exchanging integral and differentiation 7 lp-spaces and the radon-nikodym theorem 7.1 definitions 7.2 inequalities and the fischer-riesz theorem 7.3 hilbert spaces 7.4 lebesgue's decomposition theorem 7.5 supplement: signed measures 7.6 supplement: dual spaces 8 conditional expectations 8.1 elementary conditional probabilities 8.2 conditional expectations 8.3 regular conditional distribution 9 martingales 9.1 processes, filtrations, stopping times 9.2 martingales 9.3 discrete stochastic integral 9.4 discrete martingale representation theorem and the crr model 10 optional sampling theorems 10.1 doob decomposition and square variation 10.2 optional sampling and optional stopping 10.3 uniform integrability and optional sampling 11 martingale convergence theorems and their applications 11.1 doob's inequality 11.2 martingale convergence theorems 11.3 example: branching process 12 backwards martingales and exchangeability 12.1 exchangeable families of random variables 12.2 backwards martingales 12.3 de finetti's theorem 13 convergence of measures 13.1 a topology primer 13.2 weak and vague convergence 13.3 prohorov's theorem 13.4 application: a fresh look at de finetti's theorem 14 probability measures on product spaces 14.1 product spaces 14.2 finite products and transition kernels 14.3 kolmogorov's extension theorem 14.4 markov semigroups 15 characteristic functions and the central limit theorem 15.1 separating classes of functions 15.2 characteristic functions: examples 15.3 l6vy's continuity theorem 15.4 characteristic functions and moments 15.5 the central limit theorem 15.6 multidimensional central limit theorem 16 infinitely divisible distributions 16.1 l6vy-khinchin formula 16.2 stable distributions 17 markov chains 17.1 definitions and construction 17.2 discrete markov chains: examples 17.3 discrete markov processes in continuous time 17.4 discrete markov chains: recurrence and transience 17.5 application: recurrence and transience of random walks 17.6 invariant distributions 18 convergence of markov chains 18.1 periodicity of markov chains 18.2 coupling and convergence theorem 18.3 markov chain monte carlo method 18.4 speed of convergence 19 markov chains and electrical networks 19.1 harmonic functions 19.2 reversible markov chains 19.3 finite electrical networks 19.4 recurrence and transience 19.5 network reduction 19.6 random walk in a random environment 20 ergodic theory 20.1 definitions 20.2 ergodic theorems 20.3 examples 20.4 application: recurrence of random walks 20.5 mixing 21 brownian motion 21.1 continuous versions 21.2 construction and path properties 21.3 strong markov property 21.4 supplement: feller processes 21.5 construction via l2-approximation 21.6 the space c([0, ∞)) 21.7 convergence of probability measures on c([0, ∞)) 21.8 donsker's theorem 21.9 pathwise convergence of branching processes 21.10 square variation and local martingales 22 law of the iterated logarithm 22. l iterated logarithm for the brownian motion 22.2 skorohod's embedding theorem 22.3 hartman-wintner theorem 23 large deviations 23.1 cramer's theorem 23.2 large deviations principle 23.3 sanov's theorem 23.4 varadhan's lemma and free energy 24 the poisson point process 24.1 random measures 24.2 properties of the poisson point process 24.3 the poisson-dirichlet distribution 25 the it6 integral 25.1 it6 integral with respect to brownian motion 25.2 it6 integral with respect to diffusions 25.3 the it6 formula 25.4 dirichlet problem and brownian motion 25.5 recurrence and transience of brownian motion 26 stochastic differential equations 26.1 strong solutions 26.2 weak solutions and the martingale problem 26.3 weak uniqueness via duality references notation index name index subject index

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《概率論教程 》是一部講述現(xiàn)代概率論及其測度論應(yīng)用基礎(chǔ)的教程，其目標(biāo)讀者是該領(lǐng)域的研究生和相關(guān)的科研人員。內(nèi)容廣泛，有許多初級教程不能涉及到得的。理論敘述嚴(yán)謹(jǐn)，獨立性強(qiáng)。有關(guān)測度的部分和概率的章節(jié)相互交織，將概率的抽象性完全呈現(xiàn)出來。此外，還有大量的圖片、計算模擬、重要數(shù)學(xué)家的個人傳記和大量的例子。這使得表現(xiàn)形式更加活躍。本書由凱蘭克著。

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