文档介绍：Probability and Stochastic Processes
with Applications
Oliver Knill
Contents
1 Introduction 3
What is probability theory? . . . . . . . . . . . . . . . . . . 3
About these notes . . . . . . . . . . . . . . . . . . . . . . . 10
To the literature . . . . . . . . . . . . . . . . . . . . . . . . 11
Some paradoxons in probability theory . . . . . . . . . . . . 12
Some applications of probability theory . . . . . . . . . . . 15
2 Limit theorems 23
Probability spaces, random variables, independence . . . . . 23
Kolmogorov’s 0 1 law, Borel-Cantelli lemma . . . . . . . . 34
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Integration, Expectation, Variance . . . . . . . . . . . . . . 39
Results from real analysis . . . . . . . . . . . . . . . . . . . 42
Some inequalities . . . . . . . . . . . . . . . . . . . . . . . . 44
The weak law of large numbers . . . . . . . . . . . . . . . . 49
The probability distribution function . . . . . . . . . . . . . 55
Convergence of random variables . . . . . . . . . . . . . . . 58
The strong law of large numbers . . . . . . . . . . . . . . . 63
Birkhoff’s ergodic theorem . . . . . . . . . . . . . . . . . . . 67
More convergence results . . . . . . . . . . . . . . . . . . . . 71
Classes of random variables . . . . . . . . . . . . . . . . . . 77
Weak convergence . . . . . . . . . . . . . . . . . . . . . . . 89
The central limit theorem . . . . . . . . . . . . . . . . . . . 91
Entropy of distributions . . . . . . . . . . . . . . . . . . . . 97
Markov operators . . . . . . . . . . . . . . . . . . . . . . . . 106
Characteristic functions . . . . . . . . . . . . . . . . . . . . 109
The law of the iterated logarithm . . . . . . . . . . . . . . . 116
3 Discrete Stochastic Processes 121
Conditional Expectation . . . . . . . . . . . . . . . . . . . . 121
Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Doo