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A User's Guide to Measure Theoretic Probability : Cambridge Series in Statistical and Probabilistic Mathematics - David Pollard

A User's Guide to Measure Theoretic Probability

Cambridge Series in Statistical and Probabilistic Mathematics

Paperback Published: 22nd April 2002
ISBN: 9780521002899
Number Of Pages: 366

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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.

'A really useful book ...'. EMS Newsletter

Prefacep. xi
Motivation
Why bother with measure theory?p. 1
The cost and benefit of rigorp. 3
Where to start: probabilities or expectations?p. 5
The de Finetti notationp. 7
Fair pricesp. 11
Problemsp. 13
Notesp. 14
A modicum of measure theory
Measures and sigma-fieldsp. 17
Measurable functionsp. 22
Integralsp. 26
Construction of integrals from measuresp. 29
Limit theoremsp. 31
Negligible setsp. 33
L[superscript p] spacesp. 36
Uniform integrabilityp. 37
Image measures and distributionsp. 39
Generating classes of setsp. 41
Generating classes of functionsp. 43
Problemsp. 45
Notesp. 51
Densities and derivatives
Densities and absolute continuityp. 53
The Lebesgue decompositionp. 58
Distances and affinities between measuresp. 59
The classical concept of absolute continuityp. 65
Vitali covering lemmap. 68
Densities as almost sure derivativesp. 70
Problemsp. 71
Notesp. 75
Product spaces and independence
Independencep. 77
Independence of sigma-fieldsp. 80
Construction of measures on a product spacep. 83
Product measuresp. 88
Beyond sigma-finitenessp. 93
SLLN via blockingp. 95
SLLN for identically distributed summandsp. 97
Infinite product spacesp. 99
Problemsp. 102
Notesp. 108
Conditioning
Conditional distributions: the elementary casep. 111
Conditional distributions: the general casep. 113
Integration and disintegrationp. 116
Conditional densitiesp. 118
Invariancep. 121
Kolgomorov's abstract conditional expectationp. 123
Sufficiencyp. 128
Problemsp. 131
Notesp. 135
Martingale et al.
What are they?p. 138
Stopping timesp. 142
Convergence of positive supermartingalesp. 147
Convergence of submartingalesp. 151
Proof of the Krickeberg decompositionp. 152
Uniform integrabilityp. 153
Reversed martingalesp. 155
Symmetry and exchangeabilityp. 159
Problemsp. 162
Notesp. 166
Convergence in distribution
Definition and consequencesp. 169
Lindeberg's method for the central limit theoremp. 176
Multivariate limit theoremsp. 181
Stochastic order symbolsp. 182
Weakly convergent subsequencesp. 184
Problemsp. 186
Notesp. 190
Fourier transforms
Definitions and basic propertiesp. 193
Inversion formulap. 195
A mystery?p. 198
Convergence in distributionp. 198
A martingale central limit theoremp. 200
Multivariate Fourier transformsp. 202
Cramer-Wold without Fourier transformsp. 203
The Levy-Cramer theoremp. 205
Problemsp. 206
Notesp. 208
Brownian motion
Prerequisitesp. 211
Brownian motion and Wiener measurep. 213
Existence of Brownian motionp. 215
Finer properties of sample pathsp. 217
Strong Markov propertyp. 219
Martingale characterizations of Brownian motionp. 222
Functionals of Brownian motionp. 226
Option pricingp. 228
Problemsp. 230
Notesp. 234
Representations and couplings
What is coupling?p. 237
Almost sure representationsp. 239
Strassen's Theoremp. 242
The Yurinskii couplingp. 244
Quantile coupling of Binomial with normalp. 248
Haar coupling--the Hungarian constructionp. 249
The Komlos-Major-Tusnady couplingp. 252
Problemsp. 256
Notesp. 258
Exponential tails and the law of the iterated logarithm
LIL for normal summandsp. 261
LIL for bounded summandsp. 264
Kolmogorov's exponential lower boundp. 266
Identically distributed summandsp. 268
Problemsp. 271
Notesp. 272
Multivariate normal distributions
Introductionp. 274
Fernique's inequalityp. 275
Proof of Fernique's inequalityp. 276
Gaussian isoperimetric inequalityp. 278
Proof of the isoperimetric inequalityp. 280
Problemsp. 285
Notesp. 287
Measures and integrals
Measures and inner measurep. 289
Tightnessp. 291
Countable additivityp. 292
Extension to the [intersection]c-closurep. 294
Lebesgue measurep. 295
Integral representationsp. 296
Problemsp. 300
Notesp. 300
Hilbert spaces
Definitionsp. 301
Orthogonal projectionsp. 302
Orthonormal basesp. 303
Series expansions of random processesp. 305
Problemsp. 306
Notesp. 306
Convexity
Convex sets and functionsp. 307
One-sided derivativesp. 308
Integral representationsp. 310
Relative interior of a convex setp. 312
Separation of convex sets by linear functionalsp. 313
Problemsp. 315
Notesp. 316
Binomial and normal distributions
Tails of the normal distributionsp. 317
Quantile coupling of Binomial with normalp. 320
Proof of the approximation theoremp. 324
Notesp. 328
Martingales in continuous time
Filtrations, sample paths, and stopping timesp. 329
Preservation of martingale properties at stopping timesp. 332
Supermartingales from their rational skeletonsp. 334
The Brownian filtrationp. 336
Problemsp. 338
Notesp. 338
Disintegration of measures
Representation of measures on product spacesp. 339
Disintegrations with respect to a measurable mapp. 342
Problemsp. 343
Notesp. 345
Indexp. 347
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9780521002899
ISBN-10: 0521002893
Series: Cambridge Series in Statistical and Probabilistic Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 366
Published: 22nd April 2002
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 25.4 x 17.8  x 2.3
Weight (kg): 0.64

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