Yuan Shih Chow, Henry Teicher

Probability Theory (eBook, PDF)

Independence, Interchangeability, Martingales

• Format: PDF

Produktbeschreibung

Comprising the major theorems of probability theory and the measure theoretical foundations of the subject, the main topics treated here are independence, interchangeability, and martingales. Particular emphasis is placed upon stopping times, both as tools in proving theorems and as objects of interest themselves. No prior knowledge of measure theory is assumed and a unique feature of the book is the combined presentation of measure and probability. It is easily adapted for graduate students familiar with measure theory using the guidelines given.

Special features include:

- A comprehensive treatment of the law of the iterated logarithm

- The Marcinklewicz-Zygmund inequality, its extension to martingales and applications thereof

- Development and applications of the second moment analogue of Walds equation

- Limit theorems for martingale arrays; the central limit theorem for the interchangeable and martingale cases; moment convergence in the central limit theorem

- Complete discussion, including central limit theorem, of the random casting of r balls into n cells

- Recent martingale inequalities

- Cram r-L vy theorem and factor-closed families of distributions.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 489
• Erscheinungstermin: 28. November 2012
• Englisch
• ISBN-13: 9781461219507
• Artikelnr.: 44178952

Inhaltsangabe

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.
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.
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.
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.