Stationary and Related Stochastic Processes : Sample Function Properties and Their Applications

1 EMPIRICAL BACKGROUND 1(11) 1.1 Random experiments and random variables 1(2) 1.2 Finite families of random variables 3(1) 1.3 Infinite families-stochastic processes 3(4) 1.4 The probabilistic structure of a stochastic process 7(2) 1.5 Generalizations 9(3) 2 SOME FUNDAMENTAL CONCEPTS AND RESULTS OF MATHEMATICAL 12 PROBABILITY THEORY 2.1 Random experiments. Fields of events 12(2) 2.2 Events and sets. Fields and σ-fields 14(2) 2.3 Probability measure and its extensions 16(1) 2.4 Probability spaces 17(1) 2.5 Random variables 18(2) 2.6 Conditional probability. Independence 20(1) 2.7 Characteristic functions. Convergence of distributions 21(2) 2.8 Sequences of events and of random variables 23(1) 2.9 σ-fields generated by random variables 24(1) 2.10 The normal distribution 24(4) 3 FOUNDATIONS OF THE THEORY OF STOCHASTIC PROCESSES 28(34) 3.1 Families of random variables 28(2) 3.2 Sample functions 30(2) 3.3 The Kolmogorov theorem 32(5) 3.4 Real-valued parameter. The discrete case 37(2) 3.5 Convergence of sequences of random variables 39(5) 3.6 Real-valued parameter. The continuous case 44(5) 3.7 The Poisson process 49(3) 3.8 Stationary streams of events 52(6) 3.9 Examples of stochastic processes 58(4) 4 ANALYTICAL PROPERTIES OF SAMPLE FUNCTIONS 62(17) 4.1 Preliminary remarks 62(1) 4.2 Sample function continuity 63(4) 4.3 Sample function differentiability 67(3) 4.4 Questions related to continuity 70(5) 4.5 Tangencies and crossings of a level 75(4) 5 PROCESSES WITH FINITE SECOND-ORDER MOMENTS 79(30) 5.1 The mean and covariance functions 79(3) 5.2 Continuity and differentiability in quadratic mean 82(3) 5.3 Stochastic integrals in q.m. 85(3) 5.4 Analytical properties of sample functions 88(4) 5.5 Ergodic theorems 92(4) 5.6 Hilbert space and random variables 96(8) 5.7 Linear least-squares prediction 104(5) 6 PROCESSES WITH ORTHOGONAL INCREMENTS 109(11) 6.1 Generalities. Continuity in q.m. 109(3) 6.2 Sample function continuity 112(1) 6.3 Stochastic integrals 113(2) 6.4 Fourier integrals 115(3) 6.5 Processes with independent increments 118(2) 7 STATIONARY PROCESSES 120(40) 7.1 Stationary and strictly stationary processes 120(3) 7.2 The covariance function 123(2) 7.3 Analytical properties of sample functions 125(1) 7.4 Bochner's theorem 126(2) 7.5 The spectral representation 128(7) 7.6 The real-valued case 135(2) 7.7 The discrete-time case 137(1) 7.8 Linear operations on stationary processes 138(5) 7.9 Prediction 143(4) 7.10 Ergodic theorems. Stationary processes 147(1) 7.11 Ergodic theorems. Strictly stationary processes 148(12) 8 GENERALIZATIONS 160(9) 8.1 Stationary vector processes 160(3) 8.2 Processes with stationary increments 163(3) 8.3 Harmonizable processes 166(1) 8.4 Homogeneous random fields 167(2) 9 ANALYTICAL PROPERTIES OF THE SAMPLE FUNCTIONS OF NORMAL PROCESSES 169(21) 9.1 Preliminary remarks 169(1) 9.2 Stationary normal processes 170(4) 9.3 Conditions on covariance function and spectrum 174(9) 9.4 Nonstationary normal processes 183(4) 9.5 Further results 187(3) 10 "CROSSING" PROBLEMS AND RELATED TOPICS 190(29) 10.1 Background and notation 190(1) 10.2 Crossings, upcrossings, and downcrossings 191(2) 10.3 Crossing of a level-mean number in time T 193(4) 10.4 Tangencies, and the means of Nu, Uu, Du 197(3) 10.5 Crossings as stationary streams 200(2) 10.6 Higher moments of Cu, Uu, Du 202(7) 10.7 The variance of the number of zeros 209(3) 10.8 Time spent above a level; Zn-exceedance measures 212(7) 11 PROPERTIES OF STREAMS OF CROSSINGS 219(37) 11.1 Conditional distributions 219(3) 11.2 Excursions, and intervals between crossings 222(7) 11.3 Higher-order distributions 229(5) 11.4 Crossing intervals containing a fixed point 234(3) 11.5 Ergodic theorems 237(5) 11.6 Local maxima of strictly stationary processes 242(6) 11.7 The envelope of a stationary process 248(4) 11.8 Level crossings by the envelope. Fades 252(4) 12 LIMIT THEOREMS FOR CROSSINGS 256(27) 12.1 Introduction 256(1) 12.2 Asymptotic Poisson character of the stream of upcrossings 257(14) 12.3 Limiting distribution of extreme values 271(1) 12.4 Intervals between upcrossings 272(2) 12.5 Length of an excursion above a high level 274(5) 12.6 Length of a fade below a small level 279(2) 12.7 The discrete-time case 281(2) 13 NONSTATIONARY NORMAL PROCESSES. CURVE CROSSING PROBLEMS 283(16) 13.1 Curve crossings and nonstationarity 283(1) 13.2 Mean number of curve crossings 284(4) 13.3 Upcrossings, downcrossings, tangencies, and zeros 288(3) 13.4 Exceedance measures 291(3) 13.5 Extreme values of certain nonstationary normal processes 294(5) 14 FREQUENCY DETECTION AND RELATED TOPICS 299(25) 14.1 Introductory remarks 299(1) 14.2 Envelope, phase, and frequency 300(6) 14.3 Noise in frequency modulation receivers 306(8) 14.4 Sine waves in random noise 314(6) 14.5 Frequency measurements 320(4) 15 SOME ASPECTS OF THE RELIABILITY OF LINEAR SYSTEMS 324(15) 15.1 Background 324(2) 15.2 Spectral analysis of linear systems 326(5) 15.3 Application to a simple guidance system 331(8) REFERENCES 339(6) INDEX 345