Cross-covariance
In probability and statistics, given two stochastic processes { X t } {\displaystyle \left\{X_{t}\right\}} and { Y t } {\displaystyle \left\{Y_{t}\right\}} , the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation E {\displaystyle \operatorname {E} } for the expectation operator, if the processes have the mean functions μ X ( t ) = E [ X t ] {\displaystyle \mu _{X}(t)=\operatorname {\operatorname {E} } [X_{t}]} and μ Y ( t ) = E [ Y t ] {\displaystyle \mu _{Y}(t)=\operatorname {E} [Y_{t}]} , then the cross-covariance is given by K X Y ( t 1 , t 2 ) = cov ( X t 1 , Y t 2 ) = E [ ( X t 1 − μ X ( t 1 ) ) ( Y t 2 − μ Y ( t 2 ) ) ] = E [ X t 1 Y t 2 ] − μ X ( t 1 ) μ Y ( t 2 ) .