Probability Theory
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Reviews / Votes
"The book is intended for graduate students with some knowledge of measure and integration, and contains exercises. The choice of topics and its thoughtful presentation place this book among those to be seriously considered as a text for a graduate course in probability." Mathematical Reviews
"For readers having a solid background in measure and integration theory, this book is an excellent guide from basic concepts to advanced topics in modern probability theory." Zentralblatt für Mathematik
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Content
- Intro
- Preface
- Table of Contents
- Interdependence of chapters
- Notation
- Introduction
- Chapter I Basic Concepts of the Theory
- §1 Probability spaces and the language of probability theory
- §2 Laplace experiments and conditional probabilities
- §3 Random variables: Distribution, expected value, variance, Jensen's inequality
- §4 Special distributions and their properties
- §5 Convergence of random variables and distributions
- Chapter II Independence
- §6 Independent events and s-algebras
- §7 Independent random variables
- §8 Products and sums of independent random variables
- §9 Infinite products of probability spaces
- Chapter III Laws of Large Numbers
- §10 Posing the question
- §11 Zero-one laws
- §12 Strong Law of Large Numbers
- §13 Applications
- §14 Almost sure convergence of infinite series
- Chapter IV Martingales
- §15 Conditional expectations
- §16 Martingales - definition and examples
- §17 Transformation via optional times
- §18 Inequalities for supermartingales
- §19 Convergence theorems
- §20 Applications
- Chapter V Fourier Analysis
- §21 Integration of complex-valued functions
- §22 Fourier transformation and characteristic functions
- §23 Uniqueness and Continuity Theorems
- §24 Normal distribution and independence
- §25 Differentiability of Fourier transforms
- §26 Continuous mappings into the circle
- Chapter VI Limit Distributions
- §27 Examples of limit theorems
- §28 The Central Limit Theorem
- §29 Infinitely divisible distributions
- §30 Gauss measures and multi-dimensional central limit theorem
- Chapter VII Law of the Iterated Logarithm
- §31 Posing the question and elementary preparations
- §32 Probabilistic preparations
- §33 Strassen's theorem of the iterated logarithm
- §34 Supplements
- Chapter VIII Construction of Stochastic Processes
- §35 Projective limits of probability measures
- §36 Kernels and semigroups of kernels
- §37 Processes with stationary and independent increments
- §38 Processes with pre-assigned path-set
- §39 Continuous modifications
- §40 Brownian motion as a stochastic process
- §41 Poisson processes
- §42 Markov processes
- §43 Gauss processes
- §44 Conditional distributions
- Chapter IX Brownian Motion
- §45 Brownian motion with filtration and martingales
- §46 Maximal inequalities for martingales
- §47 Behavior of Brownian paths
- §48 Examples of stochastic integrals
- §49 Optional times and optional sampling
- §50 The strong Markov property
- §51 Prospectus
- Bibliography
- Symbol Index
- Name Index
- General Index
