“Probability and Stochastic Processes with Applications“ by Oliver Knill

经典的随机过程与概率入门教课书，举例丰富，可读性强
内容
1 Introduction 5
1.1 What is probability theory? . . . . . . . . . . . . . . . . . . 5
1.2 Some paradoxes in probability theory . . . . . . . . . . . . 12
1.3 Some applications of probability theory . . . . . . . . . . . 16
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 . . . . . . . . . . . . . . . . 50
2.7 The probability distribution function . . . . . . . . . . . . . 56
2.8 Convergence of random variables . . . . . . . . . . . . . . . 59
2.9 The strong law of large numbers . . . . . . . . . . . . . . . 64
2.10 Birkhoff’s ergodic theorem . . . . . . . . . . . . . . . . . . . 68
2.11 More convergence results . . . . . . . . . . . . . . . . . . . . 72
2.12 Classes of random variables . . . . . . . . . . . . . . . . . . 78
2.13 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . 90
2.14 The central limit theorem . . . . . . . . . . . . . . . . . . . 92
2.15 Entropy of distributions . . . . . . . . . . . . . . . . . . . . 98
2.16 Markov operators . . . . . . . . . . . . . . . . . . . . . . . . 107
2.17 Characteristic functions . . . . . . . . . . . . . . . . . . . . 110
2.18 The law of the iterated logarithm . . . . . . . . . . . . . . . 117
3 Discrete Stochastic Processes 123
3.1 Conditional Expectation . . . . . . . . . . . . . . . . . . . . 123
3.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3 Doob’s convergence theorem . . . . . . . . . . . . . . . . . . 143
3.4 L´evy’s upward and downward theorems . . . . . . . . . . . 150
3.5 Doob’s decomposition of a stochastic process . . . . . . . . 152
3.6 Doob’s submartingale inequality . . . . . . . . . . . . . . . 157
3.7 Doob’s Lp inequality . . . . . . . . . . . . . . . . . . . . . . 159
3.8 Random walks . . . . . . . . . . . . . . . . . . . . . . . . . 162
1
2 Contents
3.9 The arc-sin law for the 1D random walk . . . . . . . . . . . 167
3.10 The random walk on the free group . . . . . . . . . . . . . . 171
3.11 The free Laplacian on a discrete group . . . . . . . . . . . . 175
3.12 A discrete Feynman-Kac formula . . . . . . . . . . . . . . . 179
3.13 Discrete Dirichlet problem . . . . . . . . . . . . . . . . . . . 181
3.14 Markov processes . . . . . . . . . . . . . . . . . . . . . . . . 186
4 Continuous Stochastic Processes 191
4.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . 191
4.2 Some properties of Brownian motion . . . . . . . . . . . . . 198
4.3 The Wiener measure . . . . . . . . . . . . . . . . . . . . . . 205
4.4 L´evy’s modulus of continuity . . . . . . . . . . . . . . . . . 207
4.5 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . 209
4.6 Continuous time martingales . . . . . . . . . . . . . . . . . 215
4.7 Doob inequalities . . . . . . . . . . . . . . . . . . . . . . . . 217
4.8 Khintchine’s law of the iterated logarithm . . . . . . . . . . 219
4.9 The theorem of Dynkin-Hunt . . . . . . . . . . . . . . . . . 222
4.10 Self-intersection of Brownian motion . . . . . . . . . . . . . 223
4.11 Recurrence of Brownian motion . . . . . . . . . . . . . . . . 228
4.12 Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . 230
4.13 The quantum mechanical oscillator . . . . . . . . . . . . . . 235
4.14 Feynman-Kac for the oscillator . . . . . . . . . . . . . . . . 238
4.15 Neighborhood of Brownian motion . . . . . . . . . . . . . . 241
4.16 The Ito integral for Brownian motion . . . . . . . . . . . . . 245
4.17 Processes of bounded quadratic variation . . . . . . . . . . 255
4.18 The Ito integral for martingales . . . . . . . . . . . . . . . . 260
4.19 Stochastic differential equations . . . . . . . . . . . . . . . . 264
5 Selected Topics 275
5.1 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
5.2 Random Jacobi matrices . . . . . . . . . . . . . . . . . . . . 286
5.3 Estimation theory . . . . . . . . . . . . . . . . . . . . . . . 292
5.4 Vlasov dynamics . . . . . . . . . . . . . . . . . . . . . . . . 298
5.5 Multidimensional distributions . . . . . . . . . . . . . . . . 306
5.6 Poisson processes . . . . . . . . . . . . . . . . . . . . . . . . 311
5.7 Random maps . . . . . . . . . . . . . . . . . . . . . . . . . . 316
5.8 Circular random variables . . . . . . . . . . . . . . . . . . . 319
5.9 Lattice points near Brownian paths . . . . . . . . . . . . . . 327
5.10 Arithmetic random variables . . . . . . . . . . . . . . . . . 333
5.11 Symmetric Diophantine Equations . . . . . . . . . . . . . . 343
5.12 Continuity of random variables . . . . . . . . . . . . . . . . 349