A User's Guide to Measure Theoretic Probability

by David Pollard

• ISBN13: 9780521802420
• ISBN10: 0521802423
• Format: Hardcover
• Copyright: 2001-12-17
• Publisher: Cambridge University Press

Summary

Rigorous probabilistic arguments, built on the foundation of measure theory introduced seventy years ago by Kolmogorov, have invaded many fields. Students of statistics, biostatistics, econometrics, finance, and other changing disciplines now find themselves needing to absorb theory beyond what they might have learned in the typical undergraduate, calculus-based probability course. This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is 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 arguments and understand what they mean.

Author Biography

David Pollard: Yale University

Table of Contents

Preface	xi
Motivation
Why bother with measure theory?	1	(2)
The cost and benefit of rigor	3	(2)
Where to start: probabilities or expectations?	5	(2)
The de Finetti notation	7	(4)
Fair prices	11	(2)
Problems	13	(1)
Notes	14	(3)
A modicum of measure theory
Measures and sigma-fields	17	(5)
Measurable fuctions	22	(4)
Integrals	26	(3)
Construction of integrals from measures	29	(2)
Limit theorems	31	(2)
Negligible sets	33	(3)
Lp spaces	36	(1)
Uniform integrability	37	(2)
Image measures and distributions	39	(2)
Generating classes of sets	41	(2)
Generating classes of functions	43	(2)
Problems	45	(6)
Notes	51	(2)
Densities and derivatives
Densities and absolute continuity	53	(5)
The Lebesgue decomposition	58	(1)
Distances and affinities between measures	59	(6)
The classical concept of absolute continuity	65	(3)
Vitali covering lemma	68	(2)
Densities as almost sure derivatives	70	(1)
Problems	71	(4)
Notes	75	(2)
Product spaces and independence
Independence	77	(3)
Independence of sigma-fields	80	(3)
Construction of measures on a product space	83	(5)
Product measures	88	(5)
Beyond sigma-finiteness	93	(2)
SLLN via blocking	95	(2)
SLLN for identically distributed summands	97	(2)
Infinite product spaces	99	(3)
Problems	102	(6)
Notes	108	(3)
Conditioning
Conditional distributions: the elementary case	111	(2)
Conditional distributions: the general case	113	(3)
Integration and disintegration	116	(2)
Conditional densities	118	(3)
Invariance	121	(2)
Kolgomorov's abstract conditional expectation	123	(5)
Sufficiency	128	(3)
Problems	131	(4)
Notes	135	(3)
Martingale et al.
What are they?	138	(4)
Stopping times	142	(5)
Convergence of positive supermartingales	147	(4)
Convergence of submartingales	151	(1)
Proof of the Krickeberg decomposition	152	(1)
Uniform integrability	153	(2)
Reversed martingales	155	(4)
Symmetry and exchangeability	159	(3)
Problems	162	(4)
Notes	166	(3)
Convergence in distribution
Definition and consequences	169	(7)
Lindeberg's method for the central limit theorem	176	(5)
Multivariate limit theorems	181	(1)
Stochastic order symbols	182	(2)
Weakly convergent subsequences	184	(2)
Problems	186	(4)
Notes	190	(3)
Fourier transforms
Definitions and basic properties	193	(2)
Inversion formula	195	(3)
A mystery?	198	(1)
Convergence in distribution	198	(2)
A martingale central limit theorem	200	(2)
Multivariate Fourier transforms	202	(1)
Cramer-Wold without Fourier transforms	203	(2)
The Levy-Cramer theorem	205	(1)
Problems	206	(2)
Notes	208	(3)
Brownian motion
Prerequisites	211	(2)
Brownian motion and Wiener measure	213	(2)
Existence of Brownian motion	215	(2)
Finer properties of sample paths	217	(2)
Strong Markov property	219	(3)
Martingale characterizations of Brownian motion	222	(4)
Functionals of Brownian motion	226	(2)
Option pricing	228	(2)
Problems	230	(4)
Notes	234	(3)
Representations and couplings
What is coupling?	237	(2)
Almost sure representations	239	(3)
Strassen's Theorem	242	(2)
The Yurinskii coupling	244	(4)
Quantile coupling of Binomial with normal	248	(1)
Haar coupling---the Hungarian construction	249	(3)
The Komlos-Major-Tusnady coupling	252	(4)
Problems	256	(2)
Notes	258	(3)
Exponential tails and the law of the iterated logarithm
LIL for normal summands	261	(3)
LIL for bounded summands	264	(2)
Kolmogorov's exponential lower bound	266	(2)
Identically distributed summands	268	(3)
Problems	271	(1)
Notes	272	(2)
Multivariate normal distributions
Introduction	274	(1)
Fernique's inequality	275	(1)
Proof of Fernique's inequality	276	(2)
Gaussian isoperimetric inequality	278	(2)
Proof of the isoperimetric inequality	280	(5)
Problems	285	(2)
Notes	287	(2)
Appendix A: Measures and integrals
1 Measures and inner measure	289	(2)
2 Tightness	291	(1)
3 Countable additivity	292	(2)
4 Extension to the ∪c-closure	294	(1)
5 Lebesgue measure	295	(1)
6 Integral representations	296	(4)
7 Problems	300	(1)
8 Notes	300	(1)
Appendix B: Hilbert spaces
1 Definitions	301	(1)
2 Orthogonal projections	302	(1)
3 Orthonormal bases	303	(2)
4 Series expansions of random processes	305	(1)
5 Problems	306	(1)
6 Notes	306	(1)
Appendix C: Convexity
1 Convex sets and functions	307	(1)
2 One-sided derivatives	308	(2)
3 Integral representations	310	(2)
4 Relative interior of a convex set	312	(1)
5 Separation of convex sets by linear functionals	313	(2)
6 Problems	315	(1)
7 Notes	316	(1)
Appendix D: Binomial and normal distributions
1 Tails of the normal distributions	317	(3)
2 Quantile coupling of Binomial with normal	320	(4)
3 Proof of the approximation theorem	324	(4)
4 Notes	328	(1)
Appendix E: Martingales in continuous time
1 Filtrations, sample paths, and stopping times	329	(3)
2 Preservation of martingale properties at stopping times	332	(2)
3 Supermartingales from their rational skeletons	334	(2)
4 The Brownian filtration	336	(2)
5 Problems	338	(1)
6 Notes	338	(1)
Appendix F: Disintegration of measures
1 Representation of measures on product spaces	339	(3)
2 Disintegrations with respect to a measurable map	342	(1)
3 Problems	343	(2)
4 Notes	345	(2)
Index	347