Probability and measure

✍ Scribed by Patrick Billingsley

• Publisher: Wiley
• Year: 1986
• Tongue: English
• Leaves: 635
• Edition: 2 Sub
• Category: Library

✦ Synopsis

Borel's normal number theorem, proved by calculus alone, followed by short sections that establish the existence and fundamental properties of probability measures, presenting lebesque measure on the unit interval. Coverage includes key topics in measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability and stochastic processes.

✦ Table of Contents

Title......Page 1
Copyright Page......Page 2
Preface......Page 3
Contents......Page 5
1. Borel's Normal Number Theorem......Page 13
2. Probability Measures......Page 28
3. Existence and Extension......Page 44
4. Denumerable Probabilities......Page 57
5. Simple Random Variables......Page 75
6. The Law of Large Numbers......Page 92
7. Gambling Systems......Page 100
8. Markov Chains......Page 119
9. Large Deviations and the Law of the Iterated Logarithm......Page 154
10. General Measures......Page 167
11. Outer Measure......Page 174
12. Measures in Euclidean Space......Page 183
13. Measurable Functions and Mappings......Page 194
14. Distribution Functions......Page 201
15. The Integral......Page 214
16. Properties of the Integral......Page 221
17. Integral with Respect to Lebesgue Measure......Page 236
18. Product Measure and Fubini's Theorem......Page 246
19. Hausdorff Measure......Page 259
20. Random Variables and Distributions......Page 271
21. Expected Values......Page 292
22. Sums of Independent Random Variables......Page 302
23. The Poisson Process......Page 319
24. Queues and Random Walk......Page 334
25. Weak Convergence......Page 347
26. Characteristic Functions......Page 363
27. The Central Limit Theorem......Page 378
28. Infinitely Divisible Distributions......Page 394
29. Limit Theorems in R^k......Page 402
30. The Method of Moments......Page 417
31. Derivatives on the Line......Page 431
32. The Radon-Nikodym Theorem......Page 452
33. Conditional Probability......Page 460
34. Conditional Expectation......Page 478
35. Martingales......Page 492
36. Kolmogorov's Existence Theorem......Page 518
38. Separability......Page 563
APPENDIX......Page 576
NOTES ON THE PROBLEMS......Page 587
BIBLIOGRAPHY......Page 622
LIST OF SYMBOLS......Page 625
INDEX......Page 627