## An Introduction to the Theory of Stationary Random FunctionsThis two-part treatment covers the general theory of stationary random functions and the Wiener-Kolmogorov theory of extrapolation and interpolation of random sequences and processes. Beginning with the simplest concepts, it covers the correlation function, the ergodic theorem, homogenous random fields, and general rational spectral densities, among other topics. Numerous examples appear throughout the text, with emphasis on the physical meaning of mathematical concepts. Although rigorous in its treatment, this is essentially an introduction, and the sole prerequisites are a rudimentary knowledge of probability and complex variable theory. 1962 edition. |

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### Contents

THE GENERAL THEORY OF STATIONARY | 7 |

Definition of a Random Func | 16 |

Differentiation and Integration of a Random Process | 22 |

Examples | 51 |

Examples of Correlation | 61 |

FURTHER DEVELOPMENT OF THE CORRELATION THEORY OF RANDOM FUNCTIONS Page 78 | 78 |

LINEAR EXTRAPOLATION AND FILTERING | 95 |

LINEAR FILTERING OF STATIONARY RANDOM | 126 |

The Case of a General Rational Spectral | 141 |

Statement of | 150 |

The Case of a General | 161 |

LINEAR FILTERING OF STATIONARY RANDOM | 167 |

FURTHER DEVELOPMENT OF THE THEORY | 182 |

GENERALIZED RANDOM PROCESSES Page | 207 |

SOME RECENT DEVELOPMENTS Page | 214 |

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analytic function arbitrary coefficients consider constant converges correlation function correlation theory corresponding defined derivative edition entire function equations Example fact fi(T filtering problem find a function finite number Fourier function F(X given hence Hilbert space integral interval limit linear combination linear filtering Lm(t lower half-plane Markov sequences mathematical expectation mean square error mean square extrapolation mean square filtering mean value Moreover noise obtain Om(X oscillations OT(X past values polynomial prediction probability theory process E,(t process with stationary quantity random field rational function satisfy the condition sequence of uncorrelated singularities solution spectral characteristic spectral density spectral distribution function spectral representation square extrapolation error square filtering error stationary increments stationary process stationary random functions stationary random process stationary random sequence stationary sequence Stieltjes integral subspace theorem theory of stationary tion Unabridged republication uncorrelated increments uncorrelated random variables unit circle upper half-plane vanishes vectors zero