In probability and statistics, given two stochastic processes and , the crosscovariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation ; for the expectation operator, if the processes have the mean functions and , then the crosscovariance is given by
Crosscovariance is related to the more commonly used crosscorrelation of the processes in question.
In the case of two random vectors and , the crosscovariance would be a matrix (often denoted ) with entries Thus the term crosscovariance is used in order to distinguish this concept from the covariance of a random vector , which is understood to be the matrix of covariances between the scalar components of itself.
In signal processing, the crosscovariance is often called crosscorrelation and is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time between the signals, is sometimes called the sliding dot product, and has applications in pattern recognition and cryptanalysis.
The definition of crosscovariance of random vector may be generalized to stochastic processes as follows:
Let and denote stochastic processes. Then the crosscovariance function of the processes is defined by:^{[1]}^{:p.172}

where and .
If the processes are complex stochastic processes, the second factor needs to be complex conjugated.
If and are a jointly widesense stationary, then the following are true:
and
By setting (the time lag, or the amount of time by which the signal has been shifted), we may define
The crosscovariance function of two jointly WSS processes is therefore given by:

which is equivalent to
Two stochastic processes and are called uncorrelated if their covariance is zero for all times.^{[1]}^{:p.142} Formally:
The crosscovariance is also relevant in signal processing where the crosscovariance between two widesense stationary random processes can be estimated by averaging the product of samples measured from one process and samples measured from the other (and its time shifts). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or a subsampling of one of the signals). For a large number of samples, the average converges to the true covariance.
Crosscovariance may also refer to a "deterministic" crosscovariance between two signals. This consists of summing over all time indices. For example, for discretetime signals and the crosscovariance is defined as
where the line indicates that the complex conjugate is taken when the signals are complexvalued.
For continuous functions and the (deterministic) crosscovariance is defined as
The (deterministic) crosscovariance of two continuous signals is related to the convolution by
and the (deterministic) crosscovariance of two discretetime signals is related to the discrete convolution by