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Preliminaries

  • I. A. Ibragimov
  • Y. A. Rozanov
Chapter
Part of the Applications of Mathematics book series (SMAP, volume 9)

Abstract

A probability distribution P in an n-dimensional vector space ℝn is said to be Gaussian if the characteristic function
$$ \varphi (u) = \int_{{\mathbb{R}^n}} {{e^{i(u,\,x)}}P} (dx), \,\,\,\,\,\,\,\,\,u \in {\mathbb{R}^n} $$
(here (u,x)=∑u k xk denotes the scalar product of vectors u=(u1,…,un)and x = (x1,…,xn)) has the form
$$ \varphi (u) = \exp \left\{ {i(u,\,a) - \frac{1}{2}(Bu,\,u)} \right\},\,\,\,\,\,\,\,\,\,\,\,u \in {\mathbb{R}^n}, $$
(1.1)
where a = (a1…, an) ∈ ℝ is the mean and B is a linear self-adjoint non- negative definite operator called a correlation operator; the matrix {B kj }defining B is said to be a correlation matrix.

Keywords

Hilbert Space Correlation Function Scalar Product Conditional Expectation Random Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1978

Authors and Affiliations

  • I. A. Ibragimov
    • 1
  • Y. A. Rozanov
    • 2
  1. 1.LomiLeningradUSSR
  2. 2.V.A. Steklov Mathematics InstituteMoscowUSSR

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