In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If are complex-valued random variables, then the n-tuple is a complex random vector. Complex random variables can always be considered as pairs of real random vectors: their real and imaginary parts. Some concepts of real random vectors have a straightforward generalization to complex random vectors. For example, the definition of the mean of a complex random vector. Other concepts are unique to complex random vectors. Applications of complex random vectors are found in digital signal processing. Definition A complex random vector on the probability space is a function such that the vector is a real random vector on where denotes the real part of and denotes the imaginary part of . Cumulative distribution function The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form make no sense. However expressions of the form make sense. Therefore, the cumulative distribution function of a random vector is defined as where . Expectation As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise. Covariance matrix and pseudo-covariance matrix The covariance matrix (also called second central moment) contains the covariances between all pairs of components. The covariance matrix of an random vector is an matrix whose th element is the covariance between the i th and the j th random variables. Unlike in the case of real random variables, the covariance between two random variables involves the complex conjugate of one of the two. Thus the covariance matrix is a Hermitian matrix. The pseudo-covariance matrix (also called relation matrix) is defined replacing Hermitian transposition by transposition in the definition above. Properties The covariance matrix is a hermitian matrix, i.e. . The pseudo-covariance matrix is a symmetric matrix, i.e. . The covariance matrix is a positive semidefinite matrix, i.e. . Covariance matrices of real and imaginary parts By decomposing the random vector into its real part and imaginary part (i.e. ), the pair has a covariance matrix of the form: The matrices and can be related to the covariance matrices of and via the following expressions: Conversely: Cross-covariance matrix and pseudo-cross-covariance matrix The cross-covariance matrix between two complex random vectors is