Gaussian Random Processes

1. Front Matter

   Pages i-x

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2. Preliminaries

   • I. A. Ibragimov, Y. A. Rozanov

   Pages 1-27

3. The Structures of the Spaces H(T) and L _T (F)

   • I. A. Ibragimov, Y. A. Rozanov

   Pages 28-62

4. Equivalent Gaussian Distributions and their Densities

   • I. A. Ibragimov, Y. A. Rozanov

   Pages 63-107

5. Conditions for Regularity of Stationary Random Processes

   • I. A. Ibragimov, Y. A. Rozanov

   Pages 108-143

6. Complete Regularity and Processes with Discrete Time

   • I. A. Ibragimov, Y. A. Rozanov

   Pages 144-190

7. Complete Regularity and Processes with Continuous Time

   • I. A. Ibragimov, Y. A. Rozanov

   Pages 191-223

8. Filtering and Estimation of the Mean

   • I. A. Ibragimov, Y. A. Rozanov

   Pages 224-273

9. Back Matter

   Pages 274-277

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The book deals mainly with three problems involving Gaussian stationary processes. The first problem consists of clarifying the conditions for mutual absolute continuity (equivalence) of probability distributions of a "random process segment" and of finding effective formulas for densities of the equiva­ lent distributions. Our second problem is to describe the classes of spectral measures corresponding in some sense to regular stationary processes (in par­ ticular, satisfying the well-known "strong mixing condition") as well as to describe the subclasses associated with "mixing rate". The third problem involves estimation of an unknown mean value of a random process, this random process being stationary except for its mean, i. e. , it is the problem of "distinguishing a signal from stationary noise". Furthermore, we give here auxiliary information (on distributions in Hilbert spaces, properties of sam­ ple functions, theorems on functions of a complex variable, etc. ). Since 1958 many mathematicians have studied the problem of equivalence of various infinite-dimensional Gaussian distributions (detailed and sys­ tematic presentation of the basic results can be found, for instance, in [23]). In this book we have considered Gaussian stationary processes and arrived, we believe, at rather definite solutions. The second problem mentioned above is closely related with problems involving ergodic theory of Gaussian dynamic systems as well as prediction theory of stationary processes.

• Ergodic theory
• Gaussian measure
• Gaussscher Prozess
• Stationärer Prozess
• mixing
• probability measure
• random function

• Lomi, Leningrad, USSR

  I. A. Ibragimov

• V.A. Steklov Mathematics Institute, Moscow, USSR

  Y. A. Rozanov

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