Probability Theory Independence Interchangeability Martingales

Probability Theory Independence Interchangeability Martingales / by Y. S. Chow, H. Teicher.

Probability theory is a branch of mathematics dealing with chance phenomena and has clearly discernible links with the real world. The origins of the sub­ ject, generally attributed to investigations by the renowned french mathe­ matician Fermat of problems posed by a gambling contemporary to Pascal...

Full description

Saved in:
Bibliographic Details
Main Authors: Chow, Y. S. (Author), Teicher, H. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 1978.
Edition:1st ed. 1978.
Series:Springer eBook Collection.
Subjects:
Probabilities.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
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
  • 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
  • 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 expectation
  • 7.2 Conditional probabilities, conditional probability measures
  • 7.3 Conditional independence, interchangeable random variables
  • 7.4 Introduction to martingales
  • 8 Distribution Functions and Characteristic Functions
  • 8.1 Convergence of distribution functions, uniform integrability, Helly-Bray theorem
  • 8.2 Weak compactness, Frêchet-Shohat, Glivenko-Cantelli 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
  • 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 Upcrossing 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.

Similar Items