Probability Theory

1. Front Matter

   Pages i-xviii

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2. Classes of Sets, Measures, and Probability Spaces

   • Yuan Shih Chow, Henry Teicher

   Pages 1-29

3. Binomial Random Variables

   • Yuan Shih Chow, Henry Teicher

   Pages 30-53

4. Independence

   • Yuan Shih Chow, Henry Teicher

   Pages 54-83

5. Integration in a Probability Space

   • Yuan Shih Chow, Henry Teicher

   Pages 84-112

6. Sums of Independent Random Variables

   • Yuan Shih Chow, Henry Teicher

   Pages 113-158

7. Measure Extensions, Lebesgue-Stieltjes Measure Kolmogorov Consistency Theorem

   • Yuan Shih Chow, Henry Teicher

   Pages 159-201

8. Conditional Expectation, Conditional Independence, Introduction to Martingales

   • Yuan Shih Chow, Henry Teicher

   Pages 202-251

9. Distribution Functions and Characteristic Functions

   • Yuan Shih Chow, Henry Teicher

   Pages 252-294

10. Central Limit Theorems

    • Yuan Shih Chow, Henry Teicher

    Pages 295-335

11. Limit Theorems for Independent Random Variables

    • Yuan Shih Chow, Henry Teicher

    Pages 336-385

12. Martingales

    • Yuan Shih Chow, Henry Teicher

    Pages 386-423

13. Infinitely Divisible Laws

    • Yuan Shih Chow, Henry Teicher

    Pages 424-457

14. Back Matter

    Pages 458-467

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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).

• 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

• Department of Mathematical Statistics, Columbia University, New York, USA

  Yuan Shih Chow

• Department of Statistics, Rutgers University, New Brunswick, USA

  Henry Teicher

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