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Ⅰ．Introduction 3

Ⅱ．Basic Notions for Finite and Denumerable State Models 6

a．Events and Probabilities of Events 6

b．Conditional Probability，Independence，and Random Variables 10

c．The Binomial and Poisson distributions 13

d．Expectation and Variance of Random Variables (Moments) 15

e．The Weak Law of Large Numbers and the Central Limit Theorem 20

f．Entropy of an Experiment 29

g．Problems 32

Ⅲ．Markov Chains 36

a．The Markov Assumption 36

b．Matrices with Non-negative Elements (Approach of Perron-Frobenius) 44

c．Limit Properties for Markov Chains 52

d．Functions of a Markov Chain 59

e．Problems 64

Ⅳ．Probability Spaces with an Infinite Number of Sample Points 68

a．Discussion of Basic Concepts 68

b．Distribution Functions and Their Transforms 80

c．Derivatives of Measures and Conditional Probabilities 86

d．Random Processes 91

e．Problems 96

Ⅴ．Stationary Processes 100

a．Definition 100

b．The Ergodic Theorem and Stationary Processes 103

c．Convergence of Conditional Probabilities 112

d．Macmillan＇s Theorem 114

e．Problems 118

Ⅵ．Markov Processes 120

a．Definition 120

b．Jump Processes with Continuous Time 124

c．Diffusion Processes 133

d．A Refined Model of Brownian Motion 137

e．Pathological Jump Processes 141

f．Problems 146

Ⅶ．Weakly Stationary Processes and Random Harmonic Analysis 149

a．Definition 149

b．Harmonic Representation of a Stationary Process and Random Integrals 153

c．The Linear Prediction Problem and Autoregressive Schemes 160

d．Spectral Estimates for Normal Processes 169

e．Problems 178

Ⅷ．Additional Topics 182

a．A Zero-One Law 182

b．Markov Chains and Independent Random Variables 183

c．A representation for a Class of Random Processes 185

d．A Uniform Mixing Condition and Narrow Band-Pass Filtering 195

e．Problems 201

References 203

Index 207