Author | Chow, Yuan Shih. author |
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Title | Probability Theory [electronic resource] : Independence, Interchangeability, Martingales / by Yuan Shih Chow, Henry Teicher |
Imprint | New York, NY : Springer US, 1988 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0504-0 |
Descript | XVIII, 467 p. online resource |
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, Hรถlder, 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, Waldโs 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, Fubiniโs 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 -- 8 Distribution Functions and Characteristic Functions -- 8.1 Convergence of distribution functions, uniform integrability, HellyโBray theorem -- 8.2 Weak compactness, Frรฉchet-Shohat, Glivenko- Cantelli theorems -- 8.3 Characteristic functions, inversion formula, Lรฉvy continuity theorem -- 8.4 The nature of characteristic functions, analytic characteristic functions, Cramรฉr-Lรฉvy 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 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