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Apart from new examples and exercises, some simplifications of proofs, minor improvements, and correction of typographical errors, the principal change from the first edition is the addition of section 9.5, dealing with the central limit theorem for martingales and more general stochastic arrays. vii Preface to the First Edition 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, have been pushed back a century earlier to the Italian mathematicians Cardano and Tartaglia about 1570 (Ore, 1953). Results as significant as the Bernoulli weak law of large numbers appeared as early as 1713, although its counterpart, the Borel strong law oflarge numbers, did not emerge until 1909. Central limit theorems and conditional probabilities were already being investigated in the eighteenth century, but the first serious attempts to grapple with the logical foundations of probability seem to be Keynes (1921), von Mises (1928; 1931), and Kolmogorov (1933).
Content Level »Lower undergraduate
Keywords »Martingal - Martingale - Maxima - Probability theory - Random variable - conditional probability - law of the iterated logarithm - measure theory - probability space - product measure - random walk - renewal theory - uniform integrability
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 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, Helly—Bray theorem.- 8.2 Weak compactness, Fréchet-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.- 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.