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Consider a [[probability density]] [[functional]] <b>P[X(s)]</b> for stochastic variables <b>X(s)</b> at different points <b>s</b> of some space, then the correlation function is
:<math>C(s,s') = \langle X(s) X(s')\rangle</math>
where the statistical averages are taken with respect to the [[measure]] specified
by the probability density function.
In this definition, it has been assumed that the stochastic variable is a scalar. If it is not, then one can define more complicated correlation functions. For example, if one has a vector <b>X<sub>i</sub>(s)</b>, then one can define the matrix of correlation functions
:<math>C_{ij}(s,s') = \langle X_i(s) X_j(s') \rangle</math>
or a scalar, which is the trace of this matrix. If the probability density <b>P[X(s)]</b> has any target space symmetries, ie, symmetries in the space of the stochastic variable, then the correlation matrix will have induced symmetries.
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