A User's Guide to Measure Theoretic Probability

Couverture
Cambridge University Press, 10 déc. 2001
Rigorous probabilistic arguments, built on the foundation of measure theory introduced eighty 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 2002 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.
 

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Table des matières

MOTIVATION 1 Why bother with measure theory? 2 The cost and benefitofrigor 3 Where to start probabilities or expectations? 4 The de Finetti not...
CHAPTER
1 Measures and sigmafields 2 Measurable functions
3 Integrals
4 Construction of integrals from measures
5 Limit theorems
6 Negligible sets
7 Lpspaces 8 Uniform integrability
the general case
3 Integration and disintegration 4 Conditional densities
5 Invariance
6 Kolmogorovs abstract conditional expectation
7 Sufficiency 8 Problems
9 Notes
MARTINGALE ET
8 The LévyCramér theorem

9 Image measuresand distributions
10 Generatingclasses of sets
11 Generating classes of functions 12 Problems 13 Notes
DENSITIES AND DERIVATIVES 1 Densities and absolute continuity 2 The Lebesgue decomposition 3 Distances and affinities between measures
4 The classical conceptofabsolute continuity
CHAPTER CHAPTER
5 Vitali covering lemma
6 Densities as almost sure derivatives
7 Problems
PRODUCT SPACES AND INDEPENDENCE 1 Independence 2 Independence ofsigmafields 3 Constructionof measures on a product space 4 Produc...
5 Beyond sigmafiniteness
6 SLLN via blocking 7 SLLN for identically distributed summands 8 Infinite product spaces
9 Problems
10 Notes
CONDITIONING 1 Conditional distributions the elementary case
1 Whatarethey? 2 Stopping times
4 Convergence of submartingales 5 Proofofthe Krickeberg decomposition
9 Problems 10 Notes CHAPTER
EXPONENTIAL TAILSAND
CHAPTER
BINOMIAL ANDNORMAL DISTRIBUTIONS
MEASURES AND INTEGRALS
HILBERT
5 Problems 6 Notes APPENDIX
Binomial with normal
MARTINGALES IN CONTINUOUS TIME
DISINTEGRATION OF MEASURES
INDEX
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