Reference Work Entry

International Encyclopedia of Statistical Science

pp 1165-1168

Date:

Random Field

  • Mikhail P. MoklyachukAffiliated withKyiv National Taras Shevchenko University

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 tD. 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
$$\begin{array}{rcl}{ F}_{{t}_{1},\ldots,{t}_{k}}({x}_{1},\ldots,{x}_{k})& =& P\{X({t}_{1}) \\ & & <{x}_{1},\ldots,X({t}_{k}) <{x}_{k}\},k = 1,2\ldots \\ \end{array}$$
The cumulative distribution functions are by d ...
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