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.
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 .
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 .
As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise.
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 <sup>th</sup> element is the covariance between the i<sup> th</sup> and the j<sup> th</sup> 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.
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.
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:
The cross-covariance matrix between two complex random vectors is defined as:
And the pseudo-cross-covariance matrix is defined as:
Two complex random vectors and are called uncorrelated if
Two complex random vectors and are called independent if
where and denote the cumulative distribution functions of and as defined in and denotes their joint cumulative distribution function. Independence of and is often denoted by . Written component-wise, and are called independent if
A complex random vector is called circularly symmetric if for every deterministic the distribution of equals the distribution of .
A complex random vector is called proper if the following three conditions are all satisfied:
Two complex random vectors are called jointly proper if the composite random vector is proper.
The CauchyâÂÂSchwarz inequality for complex random vectors is
The characteristic function of a complex random vector with components is a function defined by: