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| Preface to the First Edition | |
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| Preface to the Second Edition | |
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| The need for measure theory | |
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| Various kinds of random variables | |
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| The uniform distribution and non-measurable sets | |
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| Exercises | |
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| Section summary | |
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| Probability triples | |
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| Basic definition | |
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| Constructing probability triples | |
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| The Extension Theorem | |
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| Constructing the Uniform[0, 1] distribution | |
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| Extensions of the Extension Theorem | |
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| Coin tossing and other measures | |
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| Exercises | |
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| Section summary | |
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| Further probabilistic foundations | |
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| Random variables | |
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| Independence | |
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| Continuity of probabilities | |
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| Limit events | |
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| Tail fields | |
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| Exercises | |
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| Section summary | |
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| Expected values | |
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| Simple random variables | |
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| General non-negative random variables | |
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| Arbitrary random variables | |
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| The integration connection | |
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| Exercises | |
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| Section summary | |
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| Inequalities and convergence | |
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| Various inequalities | |
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| Convergence of random variables | |
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| Laws of large numbers | |
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| Eliminating the moment conditions | |
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| Exercises | |
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| Section summary | |
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| Distributions of random variables | |
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| Change of variable theorem | |
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| Examples of distributions | |
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| Exercises | |
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| Section summary | |
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| Stochastic processes and gambling games | |
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| A first existence theorem | |
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| Gambling and gambler's ruin | |
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| Gambling policies | |
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| Exercises | |
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| Section summary | |
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| Discrete Markov chains | |
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| A Markov chain existence theorem | |
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| Transience, recurrence, and irreducibility | |
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| Stationary distributions and convergence | |
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| Existence of stationary distributions | |
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| Exercises | |
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| Section summary | |
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| More probability theorems | |
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| Limit theorems | |
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| Differentiation of expectation | |
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| Moment generating functions and large deviations | |
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| Fubini's Theorem and convolution | |
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| Exercises | |
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| Section summary | |
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| Weak convergence | |
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| Equivalences of weak convergence | |
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| Connections to other convergence | |
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| Exercises | |
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| Section summary | |
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| Characteristic functions | |
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| The continuity theorem | |
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| The Central Limit Theorem | |
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| Generalisations of the Central Limit Theorem | |
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| Method of moments | |
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| Exercises | |
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| Section summary | |
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| Decomposition of probability laws | |
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| Lebesgue and Hahn decompositions | |
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| Decomposition with general measures | |
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| Exercises | |
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| Section summary | |
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| Conditional probability and expectation | |
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| Conditioning on a random variable | |
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| Conditioning on a sub-[sigma]-algebra | |
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| Conditional variance | |
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| Exercises | |
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| Section summary | |
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| Martingales | |
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| Stopping times | |
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| Martingale convergence | |
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| Maximal inequality | |
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| Exercises | |
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| Section summary | |
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| General stochastic processes | |
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| Kolmogorov Existence Theorem | |
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| Markov chains on general state spaces | |
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| Continuous-time Markov processes | |
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| Brownian motion as a limit | |
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| Existence of Brownian motion | |
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| Diffusions and stochastic integrals | |
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| Ito's Lemma | |
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| The Black-Scholes equation | |
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| Section summary | |
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| Mathematical Background | |
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| Sets and functions | |
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| Countable sets | |
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| Epsilons and Limits | |
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| Infimums and supremums | |
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| Equivalence relations | |
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| Bibliography | |
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| Background in real analysis | |
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| Undergraduate-level probability | |
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| Graduate-level probability | |
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| Pure measure theory | |
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| Stochastic processes | |
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| Mathematical finance | |
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| Index | |