# Random field, homogeneous

A random field $ X ( s) $
defined on a homogeneous space $ S = \{ s \} $
of points $ s $
equipped with a transitive transformation group $ G = \{ g \} $
of mappings of $ S $
into itself, and having the property that the values of the statistical characteristics of this field do not change when elements of $ G $
are applied to their arguments. One distinguishes two different classes of homogeneous random fields: $ X ( s) $
is called a homogeneous random field in the strict sense if for all $ n = 1 , 2 \dots $
and $ g \in G $,
the finite-dimensional probability distribution of its values at any $ n $
points $ s _ {1} \dots s _ {n} $
coincides with that of its values at $ g s _ {1} \dots g s _ {n} $.
If $ {\mathsf E} | X ( s) | ^ {2} < \infty $
and $ {\mathsf E} X ( s) = {\mathsf E} X ( g s ) $,
$ {\mathsf E} X ( s) \overline{ {X ( s _ {1} ) }}\; = {\mathsf E} X ( g s ) \overline{ {X ( g s _ {1} ) }}\; $
for all $ s , s _ {1} \in S $
and $ g \in G $,
then $ X ( s) $
is called a homogeneous random field in the wide sense.

An important special case is that of a homogeneous random field on a $ k $- dimensional Euclidean space $ \mathbf R ^ {k} $( or on the lattice $ \mathbf Z ^ {k} $ of points of $ \mathbf R ^ {k} $ with integral coordinates), where $ G $ is the group of all parallel translations. Sometimes the term "homogeneous random field" is reserved for a field of this type. A homogeneous random field on $ \mathbf R ^ {k} $, with $ G $ the group of all isometric transformations of $ \mathbf R ^ {k} $( generated by parallel translations, rotations and reflections) is often called an isotropic homogeneous random field.

The concept of a homogeneous random field is a natural generalization of that of a stationary stochastic process: in both cases, the field and the covariance function admit a spectral decomposition (cf. Spectral decomposition of a random function) of special kind (see, for example, [1]–[5]). Homogeneous random fields and some of their generalizations often arise in questions of an applied nature. In particular, in the statistical theory of turbulence, an important role is played by isotropic homogeneous (scalar and vector) random fields on $ \mathbf R ^ {k} $, as well as by so-called simultaneously locally homogeneous and locally isotropic random fields (that is, fields with homogeneous and isotropic increments), which are simple generalizations of isotropic homogeneous fields (see [4], for example). Moreover, in the modern theory of physical quantum fields and in statistical physics there are applications of the theory of generalized homogeneous random fields, which include homogeneous random fields as a special case (see Random field, generalized).

#### References

[1] | A.M. Yaglom, "Second-order homogeneous random fields" , Proc. 4-th Berkeley Symp. Math. Stat. Probab. , 2 , Univ. California Press (1961) pp. 593–622 |

[2] | E.J. Hannan, "Group representations and applied probability" , Methuen (1965) |

[3] | M.I. Yadrenko, "Spectral theory of random fields" , Optim. Software (1983) (Translated from Russian) |

[4] | A.S. Monin, A.M. Yaglom, "Statistical fluid mechanics" , 2 , M.I.T. (1975) (Translated from Russian) |

[5] | A.M. Yaglom, "Correlation theory of stationary and related random functions" , 1–2 , Springer (1987) (Translated from Russian) |

#### Comments

#### References

[a1] | A.G. Ramm, "Random fields: estimation theory" , Longman & Wiley (1990) |

[a2] | H.-O. Georgii, "Gibbs measures and phase transitions" , de Gruyter (1988) |

[a3] | S. Albeverio, R. Høegh-Krohn, "Homogeneous random fields and statistical mechanics" J. Funct. Anal. , 19 (1975) pp. 242–272 |

**How to Cite This Entry:**

Random field, homogeneous.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Random_field,_homogeneous&oldid=48426