Probability: Theory and Examples Book

	Introductory Lecture	ix
1	Laws of Large Numbers	1
1.	Basic Definitions	1
2.	Random Variables	8
3.	Expected Value	12
a.	Inequalities	12
b.	Integration to the limit	15
c.	Computing expected values	17
4.	Independence	22
a.	Sufficient conditions for independence	23
b.	Independence, distribution, and expectation	26
c.	Constructing independent random variables	31
5.	Weak Laws of Large Numbers	34
a.	L[superscript 2] weak laws	34
b.	Triangular arrays	37
c.	Truncation	40
6.	Borel-Cantelli Lemmas	46
7.	Strong Law of Large Numbers	55
8.	Convergence of Random Series	60
9.	Large Deviations	69
2	Central Limit Theorems	77
1.	The De Moivre-Laplace Theorem	77
2.	Weak Convergence	80
a.	Examples	80
b.	Theory	83
3.	Characteristic Functions	89
a.	Definition, inversion formula	90
b.	Weak convergence	97
c.	Moments and derivatives	99
d.	Polya's criterion	102
e.	The moment problem	105
4.	Central Limit Theorems	110
a.	i.i.d. sequences	110
b.	Triangular arrays	114
c.	Prime divisors (Erdos-Kac)	119
d.	Rates of convergence (Berry-Esseen)	124
5.	Local Limit Theorems	129
6.	Poisson Convergence	135
a.	Basic limit theorem	135
b.	Two examples with dependence	140
c.	Poisson processes	143
7.	Stable Laws	147
8.	Infinitely Divisible Distributions	159
9.	Limit theorems in R[superscript d]	162
3	Random Walks	171
1.	Stopping Times	171
2.	Recurrence	182
3.	Visits to 0, Arcsine Laws	195
4.	Renewal Theory	202
4	Martingales	217
1.	Conditional Expectation	217
a.	Examples	219
b.	Properties	222
c.	Regular conditional probabilities	227
2.	Martingales, Almost Sure Convergence	228
3.	Examples	236
a.	Bounded increments	236
b.	Polya's urn scheme	238
c.	Radon-Nikodym derivatives	239
d.	Branching processes	243
4.	Doob's Inequality, L[superscript p] Convergence	246
	Square integrable martingales	252
5.	Uniform Integrability, Convergence in L[superscript 1]	256
6.	Backwards Martingales	262
7.	Optional Stopping Theorems	269
5	Markov Chains	274
1.	Definitions and Examples	274
2.	Extensions of the Markov Property	282
3.	Recurrence and Transience	288
4.	Stationary Measures	296
5.	Asymptotic Behavior	308
a.	Convergence theorems	308
b.	Periodic case	314
c.	Tail [sigma]-field	316
6.	General State Space	322
a.	Recurrence and transience	325
b.	Stationary measures	327
c.	Convergence theorem	328
d.	GI/G/1 queue	329
6	Ergodic Theorems	332
1.	Definitions and Examples	332
2.	Birkhoff's Ergodic Theorem	337
3.	Recurrence	343
4.	Mixing	347
5.	Entropy	353
6.	A Subadditive Ergodic Theorem	358
7.	Applications	364
7	Brownian Motion	371
1.	Definition and Construction	372
2.	Markov Property, Blumenthal's 0-1 Law	378
3.	Stopping Times, Strong Markov Property	384
4.	Maxima and Zeros	389
5.	Martingales	395
6.	Donsker's Theorem	399
7.	CLT's for Dependent Variables	408
a.	Martingales	408
b.	Stationary sequences	415
c.	Mixing properties	420
8.	Empirical Distributions, Brownian Bridge	425
9.	Laws of the Iterated Logarithm	431
Appendix	Measure Theory	437
1.	Lebesgue-Stieltjes Measures	437
2.	Caratheodory's Extension Theorem	444
3.	Completion, etc.	449
4.	Integration	452
5.	Properties of the Integral	461
6.	Product Measures, Fubini's Theorem	466
7.	Kolmogorov's Extension Theorem	471
8.	Radon-Nikodym Theorem	473
9.	Differentiating Under the Integral	478
	References	481
	Notation	489
	Normal Table	492
	Index	493