Probability Theory Independence, Interchangeability, Martingales
Now available in paperback. This is a text comprising the major theorems of probability theory and the measure theoretical foundations of the subject. The main topics treated are independence, interchangeability,and martingales; particular emphasis is placed upon stopping times, both as tools in pro...
Main Authors: | , |
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Format: | eBook |
Language: | English |
Published: |
New York, NY
Springer New York
1997, 1997
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Edition: | 3rd ed. 1997 |
Series: | Springer Texts in Statistics
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Subjects: | |
Online Access: | |
Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 1 Classes of Sets, Measures, and Probability Spaces
- 1.1 Sets and set operations
- 1.2 Spaces and indicators
- 1.3 Sigma-algebras, measurable spaces, and product spaces
- 1.4 Measurable transformations
- 1.5 Additive set functions, measures, and probability spaces
- 1.6 Induced measures and distribution functions
- 2 Binomial Random Variables
- 2.1 Poisson theorem, interchangeable events, and their limiting probabilities
- 2.2 Bernoulli, Borel theorems
- 2.3 Central limit theorem for binomial random variables, large deviations
- 3 Independence
- 3.1 Independence, random allocation of balls into cells
- 3.2 Borel-Cantelli theorem, characterization of independence, Kolmogorov zero-one law
- 3.3 Convergence in probability, almost certain convergence, and their equivalence for sums of independent random variables
- 3.4 Bernoulli trials
- 4 Integration in a Probability Space
- 4.1 Definition, properties of the integral, monotone convergence theorem
- 7.3 Conditional independence, interchangeable random variables
- 7.4 Introduction to martingales
- 7.5 U-statistics
- 8 Distribution Functions and Characteristic Functions
- 8.1 Convergence of distribution functions, uniform integrability, Helly—Bray theorem
- 8.2 Weak compactness, Fréchet—Shohat, GlivenkoCantelli theorems
- 8.3 Characteristic functions, inversion formula, Lévy continuity theorem
- 8.4 The nature of characteristic functions, analytic characteristic functions, Cramér—Lévy theorem
- 8.5 Remarks on k-dimensional distribution functions and characteristic functions
- 9 Central Limit Theorems
- 9.1 Independent components
- 9.2 Interchangeable components
- 9.3 The martingale case
- 9.4 Miscellaneous central limit theorems
- 9.5 Central limit theorems for double arrays
- 10 Limit Theorems for Independent Random Variables
- 10.1 Laws of large numbers
- 10.2 Law of the iterated logarithm
- 4.2 Indefinite integrals, uniform integrability, mean convergence
- 4.3 Jensen, Hölder, Schwarz inequalities
- 5 Sums of Independent Random Variables
- 5.1 Three series theorem
- 5.2 Laws of large numbers
- 5.3 Stopping times, copies of stopping times, Wald’s equation
- 5.4 Chung—Fuchs theorem, elementary renewal theorem, optimal stopping
- 6 Measure Extensions, Lebesgue—Stieltjes Measure,Kolmogorov Consistency Theorem
- 6.1 Measure extensions, Lebesgue—Stieltjes measure 165 6.2 Integration in a measure space
- 6.3 Product measure, Fubini’s theorem, n-dimensional Lebesgue—Stieltjes measure
- 6.4 Infinite-dimensional product measure space, Kolmogorov consistency theorem
- 6.5 Absolute continuity of measures, distribution functions; Radon—Nikodym theorem
- 7 Conditional Expectation, Conditional Independence,Introduction to Martingales
- 7.1 Conditional expectations
- 7.2 Conditional probabilities, conditional probability measures
- 10.3 Marcinkiewicz—Zygmund inequality, dominated ergodic theorems
- 10.4 Maxima of random walks
- 11 Martingales
- 11.1 Uperossing inequality and convergence
- 11.2 Martingale extension of Marcinkiewicz-Zygmund inequalities
- 11.3 Convex function inequalities for martingales
- 11.4 Stochastic inequalities
- 12 Infinitely Divisible Laws
- 12.1 Infinitely divisible characteristic functions
- 12.2 Infinitely divisible laws as limits
- 12.3 Stable laws