Preface vii
Measures and Integration: An Informal Introduction 1
1 Measures 9
1.1 Classes of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 The extension theorems and Lebesgue-Stieltjes measures . . 19
1.3.1 Caratheodory extension of measures . . . . . . . . . 19
1.3.2 Lebesgue-Stieltjes measures on R . . . . . . . . . . . 25
1.3.3 Lebesgue-Stieltjes measures on R2 . . . . . . . . . . 27
1.3.4 More on extension of measures . . . . . . . . . . . . 28
1.4 Completeness of measures . . . . . . . . . . . . . . . . . . . 30
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Integration 39
2.1 Measurable transformations . . . . . . . . . . . . . . . . . . 39
2.2 Induced measures, distribution functions . . . . . . . . . . . 44
2.2.1 Generalizations to higher dimensions . . . . . . . . . 47
2.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Riemann and Lebesgue integrals . . . . . . . . . . . . . . . 59
2.5 More on convergence . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Lp-Spaces 83
3.1 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2.1 Basic properties . . . . . . . . . . . . . . . . . . . . 89
3.2.2 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . 93
3.3 Banach and Hilbert spaces . . . . . . . . . . . . . . . . . . . 94
3.3.1 Banach spaces . . . . . . . . . . . . . . . . . . . . . 94
3.3.2 Linear transformations . . . . . . . . . . . . . . . . . 96
3.3.3 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . 97
3.3.4 Hilbert space . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 Differentiation 113
4.1 The Lebesgue-Radon-Nikodymtheorem . . . . . . . . . . . 113
4.2 Signed measures . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3 Functions of bounded variation . . . . . . . . . . . . . . . . 125
4.4 Absolutely continuous functions on R . . . . . . . . . . . . 128
4.5 Singular distributions . . . . . . . . . . . . . . . . . . . . . 133
4.5.1 Decomposition of a cdf . . . . . . . . . . . . . . . . . 133
4.5.2 Cantor ternary set . . . . . . . . . . . . . . . . . . . 134
4.5.3 Cantor ternary function . . . . . . . . . . . . . . . . 136
4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Product Measures, Convolutions, and Transforms 147
5.1 Product spaces and product measures . . . . . . . . . . . . 147
5.2 Fubini-Tonelli theorems . . . . . . . . . . . . . . . . . . . . 152
5.3 Extensions to products of higher orders . . . . . . . . . . . 157
5.4 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.4.1 Convolution of measures on


R, B(R)

. . . . . . . . 160
5.4.2 Convolution of sequences . . . . . . . . . . . . . . . 162
5.4.3 Convolution of functions in L1(R) . . . . . . . . . . 162
5.4.4 Convolution of functions and measures . . . . . . . . 164
5.5 Generating functions and Laplace transforms . . . . . . . . 164
5.6 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.7 Fourier transforms on R . . . . . . . . . . . . . . . . . . . . 173
5.8 Plancherel transform . . . . . . . . . . . . . . . . . . . . . . 178
5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6 Probability Spaces 189
6.1 Kolmogorov’s probability model . . . . . . . . . . . . . . . . 189
6.2 Randomvariables and randomvectors . . . . . . . . . . . . 191
6.3 Kolmogorov’s consistency theorem . . . . . . . . . . . . . . 199
6.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

7 Independence 2197.1 Independent events and randomvariables . . . . . . . . . . 219
7.2 Borel-Cantelli lemmas, tail σ-algebras, and Kolmogorov’s
zero-one law . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8 Laws of Large Numbers 237
8.1 Weak laws of large numbers . . . . . . . . . . . . . . . . . . 237
8.2 Strong laws of large numbers . . . . . . . . . . . . . . . . . 240
8.3 Series of independent randomvariables . . . . . . . . . . . . 249
8.4 Kolmogorov and Marcinkiewz-Zygmund SLLNs . . . . . . . 254
8.5 Renewal theory . . . . . . . . . . . . . . . . . . . . . . . . . 260
8.5.1 Definitions and basic properties . . . . . . . . . . . . 260
8.5.2 Wald’s equation . . . . . . . . . . . . . . . . . . . . 262
8.5.3 The renewal theorems . . . . . . . . . . . . . . . . . 264
8.5.4 Renewal equations . . . . . . . . . . . . . . . . . . . 266
8.5.5 Applications . . . . . . . . . . . . . . . . . . . . . . 268
8.6 Ergodic theorems . . . . . . . . . . . . . . . . . . . . . . . . 271
8.6.1 Basic definitions and examples . . . . . . . . . . . . 271
8.6.2 Birkhoff’s ergodic theorem. . . . . . . . . . . . . . . 274
8.7 Law of the iterated logarithm . . . . . . . . . . . . . . . . . 278
8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9 Convergence in Distribution 287
9.1 Definitions and basic properties . . . . . . . . . . . . . . . . 287
9.2 Vague convergence, Helly-Bray theorems, and tightness . . 291
9.3 Weak convergence on metric spaces . . . . . . . . . . . . . . 299
9.4 Skorohod’s theorem and the continuous mapping theorem . 303
9.5 Themethod of moments and themoment problem . . . . . 306
9.5.1 Convergence of moments . . . . . . . . . . . . . . . . 306
9.5.2 Themethod of moments . . . . . . . . . . . . . . . . 307
9.5.3 Themoment problem . . . . . . . . . . . . . . . . . 307
9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
10 Characteristic Functions 317
10.1 Definition and examples . . . . . . . . . . . . . . . . . . . . 317
10.2 Inversion formulas . . . . . . . . . . . . . . . . . . . . . . . 323
10.3 Levy-Cramer continuity theorem . . . . . . . . . . . . . . . 327
10.4 Extension to Rk . . . . . . . . . . . . . . . . . . . . . . . . 332
10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
11 Central Limit Theorems 343
11.1 Lindeberg-Feller theorems . . . . . . . . . . . . . . . . . . . 343
11.2 Stable distributions . . . . . . . . . . . . . . . . . . . . . . . 352
11.3 Infinitely divisible distributions . . . . . . . . . . . . . . . . 358
11.4 Refinements and extensions of the CLT . . . . . . . . . . . 361

11.4.1 The Berry-Esseen theorem . . . . . . . . . . . . . . . 361
11.4.2 Edgeworth expansions . . . . . . . . . . . . . . . . . 364
11.4.3 Large deviations . . . . . . . . . . . . . . . . . . . . 368
11.4.4 The functional central limit theorem . . . . . . . . . 372
11.4.5 Empirical process and Brownian bridge . . . . . . . 374
11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
12 Conditional Expectation and Conditional Probability 383
12.1 Conditional expectation: Definitions and examples . . . . . 383
12.2 Convergence theorems . . . . . . . . . . . . . . . . . . . . . 389
12.3 Conditional probability . . . . . . . . . . . . . . . . . . . . 392
12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13 Discrete Parameter Martingales 399
13.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . 399
13.2 Stopping times and optional stopping theorems . . . . . . . 405
13.3 Martingale convergence theorems . . . . . . . . . . . . . . . 417
13.4 Applications of martingalemethods . . . . . . . . . . . . . 424
13.4.1 Supercritical branching processes . . . . . . . . . . . 424
13.4.2 Investment sequences . . . . . . . . . . . . . . . . . 425
13.4.3 A conditional Borel-Cantelli lemma . . . . . . . . . . 425
13.4.4 Decomposition of probability measures . . . . . . . . 427
13.4.5 Kakutani’s theorem . . . . . . . . . . . . . . . . . . 429
13.4.6 de Finetti’s theorem . . . . . . . . . . . . . . . . . . 430
13.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
14 Markov Chains and MCMC 439
14.1 Markov chains: Countable state space . . . . . . . . . . . . 439
14.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 439
14.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . 440
14.1.3 Existence of aMarkov chain . . . . . . . . . . . . . . 442
14.1.4 Limit theory . . . . . . . . . . . . . . . . . . . . . . 443
14.2 Markov chains on a general state space . . . . . . . . . . . . 457
14.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . 457
14.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . 458
14.2.3 Chapman-Kolmogorov equations . . . . . . . . . . . 461
14.2.4 Harris irreducibility, recurrence, and minorization . . 462
14.2.5 Theminorization theorem . . . . . . . . . . . . . . . 464
14.2.6 The fundamental regeneration theorem . . . . . . . 465
14.2.7 Limit theory for regenerative sequences . . . . . . . 467
14.2.8 Limit theory of Harris recurrent Markov chains . . . 469
14.2.9 Markov chains on metric spaces . . . . . . . . . . . . 473
14.3 Markov chainMonte Carlo (MCMC) . . . . . . . . . . . . . 477
14.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 477
14.3.2 Metropolis-Hastings algorithm . . . . . . . . . . . . 478

14.3.3 The Gibbs sampler . . . . . . . . . . . . . . . . . . . 480
14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
15 Stochastic Processes 487
15.1 Continuous timeMarkov chains . . . . . . . . . . . . . . . . 487
15.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 487
15.1.2 Kolmogorov’s differential equations . . . . . . . . . . 488
15.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . 489
15.1.4 Limit theorems . . . . . . . . . . . . . . . . . . . . . 491
15.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . 493
15.2.1 Construction of SBM. . . . . . . . . . . . . . . . . . 493
15.2.2 Basic properties of SBM . . . . . . . . . . . . . . . . 495
15.2.3 Some related processes . . . . . . . . . . . . . . . . . 498
15.2.4 Some limit theorems . . . . . . . . . . . . . . . . . . 498
15.2.5 Some sample path properties of SBM . . . . . . . . 499
15.2.6 Brownian motion and martingales . . . . . . . . . . 501
15.2.7 Some applications . . . . . . . . . . . . . . . . . . . 502
15.2.8 The Black-Scholes formula for stock price option . . 503
15.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
16 Limit Theorems for Dependent Processes 509
16.1 A central limit theoremfor martingales . . . . . . . . . . . 509
16.2 Mixing sequences . . . . . . . . . . . . . . . . . . . . . . . . 513
16.2.1 Mixing coefficients . . . . . . . . . . . . . . . . . . . 514
16.2.2 Coupling and covariance inequalities . . . . . . . . . 516
16.3 Central limit theorems for mixing sequences . . . . . . . . . 519
16.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
17 The Bootstrap 533
17.1 The bootstrap method for independent variables . . . . . . 533
17.1.1 A description of the bootstrap method . . . . . . . . 533
17.1.2 Validity of the bootstrap: Samplemean . . . . . . . 535
17.1.3 Second order correctness of the bootstrap . . . . . . 536
17.1.4 Bootstrap for lattice distributions . . . . . . . . . . 537
17.1.5 Bootstrap for heavy tailed randomvariables . . . . . 540
17.2 Inadequacy of resampling single values under dependence . 545
17.3 Block bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . 547
17.4 Properties of theMBB . . . . . . . . . . . . . . . . . . . . . 548
17.4.1 Consistency ofMBB variance estimators . . . . . . . 549
17.4.2 Consistency ofMBB cdf estimators . . . . . . . . . . 552
17.4.3 Second order properties of theMBB . . . . . . . . . 554
17.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

18 Branching Processes 561
18.1 Bienyeme-Galton-Watson branching process . . . . . . . . . 562

18.2 BGW process: Multitype case . . . . . . . . . . . . . . . . . 564
18.3 Continuous time branching processes . . . . . . . . . . . . . 566
18.4 Embedding of Urn schemes in continuous time branching
processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
18.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
A Advanced Calculus: A Review 573
A.1 Elementary set theory . . . . . . . . . . . . . . . . . . . . . 573
A.1.1 Set operations . . . . . . . . . . . . . . . . . . . . . 574
A.1.2 The principle of induction . . . . . . . . . . . . . . . 577
A.1.3 Equivalence relations . . . . . . . . . . . . . . . . . . 577
A.2 Real numbers, continuity, differentiability, and integration . 578
A.2.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . 578
A.2.2 Sequences, series, limits, limsup, liminf . . . . . . . . 580
A.2.3 Continuity and differentiability . . . . . . . . . . . . 582
A.2.4 Riemann integration . . . . . . . . . . . . . . . . . . 584
A.3 Complex numbers, exponential and trigonometric functions 586
A.4 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 590
A.4.1 Basic definitions . . . . . . . . . . . . . . . . . . . . 590
A.4.2 Continuous functions . . . . . . . . . . . . . . . . . . 592
A.4.3 Compactness . . . . . . . . . . . . . . . . . . . . . . 592
A.4.4 Sequences of functions and uniform convergence . . 593
A.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
B List of Abbreviations and Symbols 599
B.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 599
B.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
References 603
Author Index 610
Subject Index 612