Introduction to the theory of random processes

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface to the English Translation......Page 4
Introduction......Page 6
Contents......Page 12
1. Definitions......Page 16
2. Correlation Functions (Covariance Functions)......Page 20
3. Gaussian Random Functions......Page 28
4. Oscillations with Random Parameters......Page 37
5. The Spectral Representations of the Correlation Function of a Stationary Process and of the Structural Function of a Process with Stationary Increments......Page 42
2 MEASURE THEORY......Page 55
1. Measure......Page 56
2. Measurable Functions......Page 65
3. Convergence in Measure......Page 71
4. Integrals......Page 76
5. Interchanging Limits and Integrations. Lp Spaces......Page 83
6. Absolute Continuity of Measures. Mappings......Page 91
7. Extension of Measures......Page 95
8. The Product of Two Measures......Page 108
1. Probability Spaces......Page 115
2. Construction of Probability Spaces......Page 119
3. Independence......Page 126
4. Series of Independent Random Variables......Page 133
5. Ergodic Theorems......Page 138
6. Conditional Probabilities and Conditional Mathematical Expectations......Page 149
1. Definition of a Random Function......Page 159
2. Separable Random Functions......Page 165
3. Measurable Random Functions......Page 171
4. Conditions for Nonexistence of Discontinuities of the Second Kind......Page 174
5. Continuous Random Functions......Page 184
1. Hilbert Spaces......Page 189
2. Hilbert Random Functions......Page 196
3. Stochastic Measures and Integrals......Page 205
4. Integral Representations of Random Functions......Page 215
5. Linear Transformations......Page 222
6. Physically Realizable Filters......Page 231
7. Prediction and Filtering of Stationary Processes......Page 241
8. General Theorems on the Prediction of Stationary Processes......Page 256
1. Measures Constructed from the jumps of a Process......Page 270
2. Continuous Components of a Process with Independent Increments......Page 279
3. Representation of Stochastically Continuous Processes with Independent Increments......Page 285
4. Properties of the Sample Functions of a Stochastically Continuous Process with Independent Increments......Page 289
5. Processes of Brownian Motion......Page 297
6. On the Growth of Homogeneous Processes with Independent Increments......Page 303
7 JUMP MARKOV PROCESSES......Page 312
1. Transition Probabilities......Page 313
2. Homogeneous Processes with Countably Many States......Page 317
3. Jump Processes......Page 326
4. Examples......Page 334
5. Branching Processes......Page 342
6. The General Definition of a Markov Process......Page 358
7. The Basic Properties of Jump Processes......Page 362
8 DIFFUSION PROCESSES......Page 385
1. Diffusion Processes in the Broad Sense......Page 387
2. Ito's Stochastic Integral......Page 393
3. Existence and Uniqueness of Solutions of Stochastic Differential Equations......Page 406
4. Differentiability of Solutions of Stochastic Equations with Respect to Initial Conditions......Page 418
5. The Method of Differential Equations......Page 427
6. One-Dimensional Diffusion Processes with Absorption......Page 435
9 LIMIT THEOREMS FOR RANDOM PROCESSES......Page 453
1. Weak Convergence of Distributions in a Metric Space......Page 455
2. Limit Theorems for Continuous Processes......Page 463
3. Convergence of Sequences of Sums of Independent Random Variables to Processes of Brownian Motion......Page 467
4. Convergence of a Sequence of Markov Chains to a Diffusion Process......Page 474
5. The Space of Functions without Discontinuities of the Second Kind......Page 484
6. Convergence of a Sequence of Sums of Identically Distributed Independent Random Variables to a Homogeneous Process with Independent Increments......Page 493
7. Limit theorems for Functionals of Integral Form......Page 499
8. Application of Limit Theorems to Statistical Criteria......Page 505
BIBLIOGRAPHIC NOTES......Page 512
BIBLIOGRAPHY......Page 518
INDEX OF SYMBOLS......Page 526
INDEX......Page 528