User's Guide to Measure Theoretic Probability

ISBN-10: 0521002893
ISBN-13: 9780521002899
Edition: 2002
List price: $57.00 Buy it from $34.43
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Description: This text is not just a presentation of mathematical theory, but also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguements and understand what  More...

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Book details

List price: $57.00
Copyright year: 2002
Publisher: Cambridge University Press
Publication date: 12/10/2001
Binding: Paperback
Pages: 366
Size: 7.00" wide x 9.75" long x 0.75" tall
Weight: 1.430
Language: English

This text is not just a presentation of mathematical theory, but also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguements and understand what they mean.

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

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