Harvard 大学 概率论、随机过程及应用讲义 By Professor Oliver Knill
Probability and Stochastic Processes with Applications
350页讲义 从概率论的最基本概念讲起 适合自学

Contents
1 Introduction 3
1.1 What is probability theory? . . . . . . . . . . . . . . . . . . 3
1.2 About these notes . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 To the literature . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Some paradoxons in probability theory . . . . . . . . . . . . 12
1.5 Some applications of probability theory . . . . . . . . . . . 15

2 Limit theorems 23
2.1 Probability spaces, random variables, independence . . . . . 23
2.2 Kolmogorov’s 0 − 1 law, Borel-Cantelli lemma . . . . . . . . 34
2.3 Integration, Expectation, Variance . . . . . . . . . . . . . . 39
2.4 Results from real analysis . . . . . . . . . . . . . . . . . . . 42
2.5 Some inequalities . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 The weak law of large numbers . . . . . . . . . . . . . . . . 49
2.7 The probability distribution function . . . . . . . . . . . . . 55
2.8 Convergence of random variables . . . . . . . . . . . . . . . 58
2.9 The strong law of large numbers . . . . . . . . . . . . . . . 63
2.10 Birkhoff’s ergodic theorem . . . . . . . . . . . . . . . . . . . 67
2.11 More convergence results . . . . . . . . . . . . . . . . . . . . 71
2.12 Classes of random variables . . . . . . . . . . . . . . . . . . 77
2.13 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . 89
2.14 The central limit theorem . . . . . . . . . . . . . . . . . . . 91
2.15 Entropy of distributions . . . . . . . . . . . . . . . . . . . . 97
2.16 Markov operators . . . . . . . . . . . . . . . . . . . . . . . . 106
2.17 Characteristic functions . . . . . . . . . . . . . . . . . . . . 109
2.18 The law of the iterated logarithm . . . . . . . . . . . . . . . 116

3 Discrete Stochastic Processes 121
3.1 Conditional Expectation . . . . . . . . . . . . . . . . . . . . 121
3.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.3 Doob’s convergence theorem . . . . . . . . . . . . . . . . . . 141
3.4 L´evy’s upward and downward theorems . . . . . . . . . . . 148
3.5 Doob’s decomposition of a stochastic process . . . . . . . . 150
3.6 Doob’s submartingale inequality . . . . . . . . . . . . . . . 155
3.7 Doob’s Lp inequality . . . . . . . . . . . . . . . . . . . . . . 157
3.8 Random walks . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.9 The arc-sin law for the 1D random walk . . . . . . . . . . . 165
3.10 The random walk on the free group . . . . . . . . . . . . . . 169
3.11 The free Laplacian on a discrete group . . . . . . . . . . . . 173
3.12 A discrete Feynman-Kac formula . . . . . . . . . . . . . . . 177
3.13 Discrete Dirichlet problem . . . . . . . . . . . . . . . . . . . 179
3.14 Markov processes . . . . . . . . . . . . . . . . . . . . . . . . 184

4 Continuous Stochastic Processes 189
4.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . 189
4.2 Some properties of Brownian motion . . . . . . . . . . . . . 196
4.3 The Wiener measure . . . . . . . . . . . . . . . . . . . . . . 203
4.4 L´evy’s modulus of continuity . . . . . . . . . . . . . . . . . 205
4.5 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . 207
4.6 Continuous time martingales . . . . . . . . . . . . . . . . . 213
4.7 Doob inequalities . . . . . . . . . . . . . . . . . . . . . . . . 215
4.8 Kintchine’s law of the iterated logarithm . . . . . . . . . . . 217
4.9 The theorem of Dynkin-Hunt . . . . . . . . . . . . . . . . . 220
4.10 Self-intersection of Brownian motion . . . . . . . . . . . . . 221
4.11 Recurrence of Brownian motion . . . . . . . . . . . . . . . . 226
4.12 Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . 228
4.13 The quantum mechanical oscillator . . . . . . . . . . . . . . 233
4.14 Feynman-Kac for the oscillator . . . . . . . . . . . . . . . . 236
4.15 Neighborhood of Brownian motion . . . . . . . . . . . . . . 239
4.16 The Ito integral for Brownian motion . . . . . . . . . . . . . 243
4.17 Processes of bounded quadratic variation . . . . . . . . . . 253
4.18 The Ito integral for martingales . . . . . . . . . . . . . . . . 258
4.19 Stochastic differential equations . . . . . . . . . . . . . . . . 262

5 Selected Topics 273
5.1 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
5.2 Random Jacobi matrices . . . . . . . . . . . . . . . . . . . . 284
5.3 Estimation theory . . . . . . . . . . . . . . . . . . . . . . . 290
5.4 Vlasov dynamics . . . . . . . . . . . . . . . . . . . . . . . . 296
5.5 Multidimensional distributions . . . . . . . . . . . . . . . . 304
5.6 Poisson processes . . . . . . . . . . . . . . . . . . . . . . . . 309
5.7 Random maps . . . . . . . . . . . . . . . . . . . . . . . . . . 314
5.8 Circular random variables . . . . . . . . . . . . . . . . . . . 317
5.9 Lattice points near Brownian paths . . . . . . . . . . . . . . 325
5.10 Arithmetic random variables . . . . . . . . . . . . . . . . . 331
5.11 Symmetric Diophantine Equations . . . . . . . . . . . . . . 341
5.12 Continuity of random variables . . . . . . . . . . . . . . . . 347