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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1957, Volume 2, Issue 1, Pages 60–91 (Mi tvp4958)

On the Linear Extrapolation of a Continuous Homogeneous Random Field

Chiang Tse-Pei

Pekin

Abstract: An homogeneous random field is defined to be a family of random variables $x(s,t)$, $-\infty<s<\infty,-\infty<t< \infty$, such that
$$\mathbf M[x({s,t})]\equiv0,\lim_{\substack{h_1\to 0 h_2\to0}}\mathbf M[x(s+h_1,t+h_2)-x(s,t)]^2=0$$
and that $B_x(s,t)=\mathbf M[x(s+m,t+n)x\overline{(m,n)}]$ exists and is independent of $m$ and $n$.
In the present paper, the extrapolation problem consists in determining the best linear prediction for the value of the field at any point in the upper half-plane by values at points in the entire lower half-plane. Various types of homogeneous random fields (singular fields, regular fields and fields of Markov type) are considered from the point of view of extrapolation theory. Necessary and sufficient conditions for these types of random fields are given in terms of spectral functions of the field. An expression for the mean square error of the optimum prediction is also obtained.

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English version:
Theory of Probability and its Applications, 1957, 2:1, 58–89

Citation: Chiang Tse-Pei, “On the Linear Extrapolation of a Continuous Homogeneous Random Field”, Teor. Veroyatnost. i Primenen., 2:1 (1957), 60–91; Theory Probab. Appl., 2:1 (1957), 58–89

Citation in format AMSBIB
\Bibitem{Chi57} \by Chiang~Tse-Pei \paper On the Linear Extrapolation of a Continuous Homogeneous Random Field \jour Teor. Veroyatnost. i Primenen. \yr 1957 \vol 2 \issue 1 \pages 60--91 \mathnet{http://mi.mathnet.ru/tvp4958} \transl \jour Theory Probab. Appl. \yr 1957 \vol 2 \issue 1 \pages 58--89 \crossref{https://doi.org/10.1137/1102003} 

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This publication is cited in the following articles:
1. Yu. A. Rozanov, “On the theory of homogeneous random fields”, Math. USSR-Sb., 32:1 (1977), 1–18
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