Measure Theory and Probability Theory

Measure Theory and Probability Theory......Page 1
Preface......Page 7
Contents......Page 12
Measures and Integration: An Informal Introduction......Page 18
1.1 Classes of sets......Page 25
1.2 Measures......Page 30
1.3.1 Caratheodory extension of measures......Page 35
1.3.2 Lebesque-Stieltjes measures on R......Page 41
1.3.3 Lebesque-Stieltjes measures on R2......Page 43
1.3.4 More on extension of measures......Page 44
1.4 Completeness of measures......Page 46
1.5 Problems......Page 47
2.1 Measureable transformations......Page 55
2.2 Induced measures, distribution functions......Page 60
2.2.1 Generalizations to higher dimensions......Page 63
2.3 Integration......Page 64
2.4 Riemann and Lebesque integrals......Page 75
2.5 More on convergence......Page 77
2.6 Problems......Page 87
3.1 Inequalities......Page 99
3.2.1 Basic properties......Page 105
3.2.2 Dual spaces......Page 109
3.3.1 Banach spaces......Page 110
3.3.2 Linear transformations......Page 112
3.3.3 Dual spaces......Page 113
3.3.4 Hilbert spaces......Page 114
3.4 Problems......Page 118
4.1 The Lebesque-Radon-Nikodym Theorem......Page 128
4.2 Signed measures......Page 134
4.3 functions of bounded variation......Page 140
4.4 Absolutely continuous functions on R......Page 143
4.5.1 Decomposition of a cdf......Page 148
4.5.2 Cantor ternary set......Page 149
4.5.3 Cantor ternary function......Page 151
4.6 Problems......Page 152
5.1 Product spaces and product measures......Page 161
5.2 Fubini-Tonelli theorems......Page 166
5.3 Extensions to products of higher orders......Page 171
5.4.1 Convolution of measures on (R, B(R))......Page 174
5.4.3 Convolution of functions in L1(R)......Page 176
5.5 Generating functions and Laplace transforms......Page 178
5.6 Fourier series......Page 180
5.7 Fourier transform on R......Page 187
5.8 Plancherel transform......Page 192
5.9 Problems......Page 195
6.1 Kolmogorov\'s probability model......Page 203
6.2 Random variables and random vectors......Page 205
6.3 Kolmogorov\'s consistency theorem......Page 213
6.4 Problems......Page 226
7.1 Independent events and random variables......Page 233
7.2 Borel-Cantelli lemmas, tail σ-algebras, and Kolmogorov\'s zero-one law......Page 236
7.3 Problems......Page 241
8.1 Weak laws of large numbers......Page 250
8.2 Strong laws of large numbers......Page 253
8.3 Series of independent random variables......Page 262
8.4 Kolmogorov and Marcinkiewz-Zygmund SLLNs......Page 267
8.5.1 Definitions and basic properties......Page 273
8.5.2 Wald\'s equations......Page 275
8.5.3 The reneval theorems......Page 277
8.5.4 Reneval equations......Page 279
8.5.5 Applications......Page 281
8.6.1 Basic definitions and examples......Page 284
8.6.2 Birkhoff\'s ergodic theorem......Page 287
8.7 Law of the iterated logarithm......Page 291
8.8 Problems......Page 292
9.1 Definitions and basic properties......Page 299
9.2 Vague convergence, Helly-Bray theorems, and tightness......Page 303
9.3 Weak convergence on metric spaces......Page 311
9.4 Skorohod\'s theorem and the continuous mapping theorem......Page 315
9.5 The method of moments and the moment problem......Page 318
9.6 Problems......Page 321
10.1 Definition and examples......Page 328
10.2 Inversion formulas......Page 334
10.3 Levy-Cramer continuity theorem......Page 338
10.4 Extension to Rk......Page 343
10.5 Problems......Page 348
11.1 Lindeberg-Feller theorems......Page 353
11.2 Stable distributions......Page 362
11.3 Infinitely divisible distributions......Page 368
11.4 Refinements and extensions of the CLT......Page 371
11.5 Problems......Page 386
12.1 Conditional expectation: Definitions and examples......Page 393
12.2 Convergence theorems......Page 399
12.3 Conditional probability......Page 402
12.4 Problems......Page 403
13.1 Definitions and examples......Page 409
13.2 Stopping times and optional stopping theorems......Page 415
13.3 Martingale convergence theorems......Page 427
13.4 Applications of martingale methods......Page 434
13.5 Problems......Page 440
14.1 Markov chains: Countable state space......Page 448
14.2 Markov chains on a general state space......Page 466
14.3 Markov chain Monte Carlo (MCMC)......Page 486
14.4 Problems......Page 490
15.1 Continuous time Markov chains......Page 496
15.2 Brownian motion......Page 502
15.3 Problems......Page 513
16.1 A central limit theorem for martingales......Page 517
16.2 Mixing sequences......Page 521
16.3 Central limit theorem for mixing sequences......Page 527
16.4 Problems......Page 537
17.1 The bootstrap methods for independent variables......Page 540
17.2 Inadequancy of resampling single values under dependence......Page 552
17.3 Block bootstrap......Page 554
17.4 Properties of the MBB......Page 555
17.5 Problems......Page 563
18 - Branching Processes......Page 568
18.1 Bienyeme-Galton-Watson branching process......Page 569
18.2 BGW process: Multitype case......Page 571
18.3 Continuous time branching processes......Page 573
18.4 Embedding of Urn schemes in continuous time branching processes......Page 575
18.5 Problems......Page 576
A.1 Elementery set theory......Page 579
A.2 Real numbers, continuity, differentiability, and integration......Page 584
A.3 Complex numbers, exponential and trigonometric functions......Page 592
A.4 Metric spaces......Page 596
B.1 Abbrevations......Page 605
B.2 Symbols......Page 606
References......Page 609
Author Index......Page 616
Subject Index......Page 618