Probability Theory: Independence, Interchangeability, Martingales

✍ Scribed by Yuan Shih Chow, Henry Teicher (auth.)

• Publisher: Springer-Verlag New York
• Year: 1988
• Tongue: English
• Leaves: 482
• Series: Springer Texts in Statistics
• Edition: 2
• Category: Library

✦ Synopsis

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).

✦ Table of Contents

Front Matter....Pages i-xviii
Classes of Sets, Measures, and Probability Spaces....Pages 1-29
Binomial Random Variables....Pages 30-53
Independence....Pages 54-83
Integration in a Probability Space....Pages 84-112
Sums of Independent Random Variables....Pages 113-158
Measure Extensions, Lebesgue-Stieltjes Measure Kolmogorov Consistency Theorem....Pages 159-201
Conditional Expectation, Conditional Independence, Introduction to Martingales....Pages 202-251
Distribution Functions and Characteristic Functions....Pages 252-294
Central Limit Theorems....Pages 295-335
Limit Theorems for Independent Random Variables....Pages 336-385
Martingales....Pages 386-423
Infinitely Divisible Laws....Pages 424-457
Back Matter....Pages 458-467

✦ Subjects

Probability Theory and Stochastic Processes