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Preface | |
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Motivation | |
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Why bother with measure theory? | |
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The cost and benefit of rigor | |
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Where to start: probabilities or expectations? | |
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The de Finetti notation | |
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Fair prices | |
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Problems | |
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Notes | |
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A modicum of measure theory | |
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Measures and sigma-fields | |
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Measurable functions | |
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Integrals | |
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Construction of integrals from measures | |
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Limit theorems | |
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Negligible sets | |
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L[superscript p] spaces | |
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Uniform integrability | |
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Image measures and distributions | |
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Generating classes of sets | |
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Generating classes of functions | |
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Problems | |
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Notes | |
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Densities and derivatives | |
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Densities and absolute continuity | |
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The Lebesgue decomposition | |
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Distances and affinities between measures | |
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The classical concept of absolute continuity | |
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Vitali covering lemma | |
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Densities as almost sure derivatives | |
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Problems | |
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Notes | |
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Product spaces and independence | |
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Independence | |
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Independence of sigma-fields | |
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Construction of measures on a product space | |
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Product measures | |
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Beyond sigma-finiteness | |
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SLLN via blocking | |
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SLLN for identically distributed summands | |
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Infinite product spaces | |
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Problems | |
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Notes | |
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Conditioning | |
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Conditional distributions: the elementary case | |
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Conditional distributions: the general case | |
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Integration and disintegration | |
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Conditional densities | |
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Invariance | |
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Kolgomorov's abstract conditional expectation | |
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Sufficiency | |
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Problems | |
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Notes | |
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Martingale et al. | |
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What are they? | |
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Stopping times | |
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Convergence of positive supermartingales | |
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Convergence of submartingales | |
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Proof of the Krickeberg decomposition | |
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Uniform integrability | |
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Reversed martingales | |
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Symmetry and exchangeability | |
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Problems | |
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Notes | |
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Convergence in distribution | |
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Definition and consequences | |
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Lindeberg's method for the central limit theorem | |
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Multivariate limit theorems | |
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Stochastic order symbols | |
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Weakly convergent subsequences | |
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Problems | |
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Notes | |
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Fourier transforms | |
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Definitions and basic properties | |
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Inversion formula | |
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A mystery? | |
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Convergence in distribution | |
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A martingale central limit theorem | |
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Multivariate Fourier transforms | |
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Cramer-Wold without Fourier transforms | |
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The Levy-Cramer theorem | |
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Problems | |
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Notes | |
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Brownian motion | |
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Prerequisites | |
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Brownian motion and Wiener measure | |
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Existence of Brownian motion | |
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Finer properties of sample paths | |
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Strong Markov property | |
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Martingale characterizations of Brownian motion | |
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Functionals of Brownian motion | |
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Option pricing | |
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Problems | |
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Notes | |
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Representations and couplings | |
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What is coupling? | |
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Almost sure representations | |
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Strassen's Theorem | |
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The Yurinskii coupling | |
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Quantile coupling of Binomial with normal | |
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Haar coupling--the Hungarian construction | |
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The Komlos-Major-Tusnady coupling | |
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Problems | |
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Notes | |
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Exponential tails and the law of the iterated logarithm | |
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LIL for normal summands | |
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LIL for bounded summands | |
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Kolmogorov's exponential lower bound | |
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Identically distributed summands | |
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Problems | |
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Notes | |
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Multivariate normal distributions | |
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Introduction | |
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Fernique's inequality | |
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Proof of Fernique's inequality | |
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Gaussian isoperimetric inequality | |
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Proof of the isoperimetric inequality | |
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Problems | |
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Notes | |
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Measures and integrals | |
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Measures and inner measure | |
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Tightness | |
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Countable additivity | |
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Extension to the [intersection]c-closure | |
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Lebesgue measure | |
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Integral representations | |
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Problems | |
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Notes | |
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Hilbert spaces | |
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Definitions | |
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Orthogonal projections | |
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Orthonormal bases | |
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Series expansions of random processes | |
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Problems | |
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Notes | |
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Convexity | |
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Convex sets and functions | |
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One-sided derivatives | |
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Integral representations | |
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Relative interior of a convex set | |
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Separation of convex sets by linear functionals | |
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Problems | |
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Notes | |
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Binomial and normal distributions | |
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Tails of the normal distributions | |
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Quantile coupling of Binomial with normal | |
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Proof of the approximation theorem | |
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Notes | |
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Martingales in continuous time | |
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Filtrations, sample paths, and stopping times | |
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Preservation of martingale properties at stopping times | |
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Supermartingales from their rational skeletons | |
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The Brownian filtration | |
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Problems | |
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Notes | |
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Disintegration of measures | |
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Representation of measures on product spaces | |
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Disintegrations with respect to a measurable map | |
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Problems | |
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Notes | |
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Index | |