Correlation Function - Definition

Definition

For random variables X(s) and X(t) at different points s and t of some space, the correlation function is

where is described in the article on correlation. In this definition, it has been assumed that the stochastic variable is scalar-valued. If it is not, then more complicated correlation functions can be defined. For example, if one has a vector X_i(s), then one can define the matrix of correlation functions

or a scalar, which is the trace of this matrix. If the probability distribution has any target space symmetries, i.e. symmetries in the space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. If there are symmetries of the space (or time) in which the random variables exist (also called spacetime symmetries) then the correlation matrix will have special properties. Examples of important spacetime symmetries are —

• translational symmetry yields C(s,s') = C(s − s') where s and s' are to be interpreted as vectors giving coordinates of the points
• rotational symmetry in addition to the above gives C(s, s') = C(|s − s'|) where |x| denotes the norm of the vector x (for actual rotations this is the Euclidean or 2-norm).

n is

If the random variable has only one component, then the indices are redundant. If there are symmetries, then the correlation function can be broken up into irreducible representations of the symmetries — both internal and spacetime.

The case of correlations of a single random variable can be thought of as a special case of autocorrelation of a stochastic process on a space which contains a single point.