Definition
For possibly distinct random variables ''X''(''s'') and ''Y''(''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 variables are scalar-valued. If they are not, then more complicated correlation functions can be defined. For example, if ''X''(''s'') is a random vector with ''n'' elements and ''Y''(t) is a vector with ''q'' elements, then an ''n''×''q'' matrix of correlation functions is defined with element : When ''n''=''q'', sometimes the trace of this matrix is focused on. If the probability distributions have any target space symmetries, i.e. symmetries in the value space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. Similarly, if there are symmetries of the space (or time) domain in which the random variables exist (also called spacetime symmetries), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are — *translational symmetry yields ''C''(''s'',''s''Properties of probability distributions
With these definitions, the study of correlation functions is similar to the study of probability distributions. Many stochastic processes can be completely characterized by their correlation functions; the most notable example is the class of Gaussian processes. Probability distributions defined on a finite number of points can always be normalized, but when these are defined over continuous spaces, then extra care is called for. The study of such distributions started with the study of random walks and led to the notion of the Itō calculus. The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics. Any probability distribution which obeys a condition on correlation functions called reflection positivity leads to a local quantum field theory after Wick rotation to Minkowski spacetime (see Osterwalder-Schrader axioms). The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory.See also
* * * * * * * * * * *References
{{Authority control Covariance and correlation Time series Spatial analysis