Basic Principles and Applications of Probability Theory (eBook)

Contents 5
Probability. Basic Notions. Structure. Methods 6
Contents 6
1 Introduction 10
1.1 The Nature of Randomness 10
1.2 Formalization of Randomness 14
1.3 Problems of Probability Theory 19
2 Probability Space 23
2.1 Finite Probability Space 23
2.2 Definition of Probability Space 31
2.3 Random Mappings 42
2.4 Construction of Probability Spaces 50
3 Independence 57
3.1 Independence of s-Algebras 57
3.2 Sequences of Independent Random Variables 63
3.3 Random Walks 71
3.4 Processes with Independent Increments 82
3.5 Product Measures 90
4 General Theory of Stochastic Processes and Random Functions 96
4.1 Regular Modifications 96
4.2 Measurability 103
4.3 Adapted Processes 107
4.4 Martingales 113
4.5 Stochastic Integrals and Integral Representations of Random Functions 118
5 Limit Theorems 122
5.1 Weak Convergence of Distributions 122
5.2 Ergodic Theorems 127
5.3 Central Limit Theorem and Invariance Principle 135
Historic and Bibliographic Comments 142
References 144
Markov Processes and Probability Applications in Analysis 146
Contents 146
Introduction 147
1 Markov Processes 148
1.1 Definition and General Properties 148
1.2 Purely Discontinuous Processes 154
1.3 Diffusion Processes 162
2 Probability Representation of Solutions of Partial Differential Equations 166
2.1 Problems for a Parabolic Equation 166
2.2 Boundary-Value Problems for Elliptic Operators 172
2.3 Wiener Measure and the Solution of Equations Involving the Laplace Operator 177
Historic and Bibliographic Comments 188
References 189
Applied Probability 190
Contents 190
Introduction 192
1 Statistical Methods 193
1.1 Processing of Statistical Data 193
1.2 Testing of Hypotheses 200
1.3 Decision-Making Under Uncertainty 205
2 Controlled Stochastic Processes 212
2.1 Controlled Random Sequences 212
2.2 Controlled Markov Chains 220
2.3 Time-Continuous Controlled Markov Processes 225
3 Information 232
3.1 Entropy 232
3.2 Transmission of Information 241
3.3 Shannon’s Theorem 246
4 Filtering 253
4.1 Linear Prediction and Filtering of Stationary Stochastic Processes 253
4.2 Nonlinear Filtering 262
Historic and Bibliographic Comments 269
Author Index 271
Subject Index 272