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*'''rotational symmetry''' in addition to the above gives ''C''(''s'', ''s''<nowiki>'</nowiki>) = ''C''(|''s'' − ''s''<nowiki>'</nowiki>|) where |''x''| denotes the norm of the vector ''x'' (for actual rotations this is the Euclidean or 2-norm).
Higher order correlation functions are often defined. A typical correlation function of order ''n'' is (the angle brackets represent the [[expectation value]]).
:<math>C_{i_1i_2\cdots i_n}(s_1,s_2,\cdots,s_n) = \langle X_{i_1}(s_1) X_{i_2}(s_2) \cdots X_{i_n}(s_n)\rangle
If the random vector has only one component variable, then the indices <math>i,j</math> are redundant. If there are symmetries, then the correlation function can be broken up into [[irreducible representation]]s of the symmetries — both internal and spacetime.
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