%%% Title of object: Random field
%%% Canonical Name: RandomField6
%%% Type: Definition
%%% Created on: 2010-10-01 11:06:22
%%% Modified on: 2010-10-01 11:06:22
%%% Creator: Moklyachuk
%%% Modifier: Moklyachuk
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%%% Classification: msc:62M40, msc:60G60
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\centerline{\bf Random field}
\centerline{Mikhail Moklyachuk}
\centerline{Department of Probability Theory, Statistics and
Actuarial Mathematics,}
\centerline{Kyiv National Taras
Shevchenko University, Ukraine}
\centerline{Email: mmp\@ univ.kiev.ua}
Keywords and Phrases: Random field; Kolmogorov Existence Theorem;
Gaussian random field; Wiener sheet; Brownian sheet; Poisson random
field; Markov random field; Homogeneous random field; Isotropic
random field; Spectral decomposition
{\it Random field} $X(t)$ on $D\subset\mathbb R^n$ (i.e. $t\in D\subset\mathbb R^n$) is a function whose values are random variables for any $t\in D$.
The dimension of the coordinate is usually in the range from one to four, but any $n>0$ is possible. A one-dimensional random field is usually called a stochastic process. The term 'random field' is used to stress that the dimension of the coordinate is higher than one. Random fields in two and three dimensions are encountered in a wide range of sciences and especially in the earth sciences such as hydrology, agriculture, and geology. Random fields where $t$ is a position in space-time are studied in turbulence theory and in meteorology.
Random field $X(t)$ is described by its finite-dimensional
(cumulative) distributions
$$F_{t_1,\dots,t_k}(x_1,\dots,x_k)=P\{X(t_1)