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.
are complex-valued random variables, then the n-tuple
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.
is a real random vector on
denotes the real part of
denotes the imaginary part of
292 The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form
Therefore, the cumulative distribution function
As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise.[1]: p.
293 The covariance matrix (also called second central moment)
contains the covariances between all pairs of components.
th element is the covariance between the i th and the j th random variables.
[2]: p.372 Unlike in the case of real random variables, the covariance between two random variables involves the complex conjugate of one of the two.
The pseudo-covariance matrix (also called relation matrix) is defined replacing Hermitian transposition by transposition in the definition above.
The covariance matrix is a positive semidefinite matrix, i.e. By decomposing the random vector
has a covariance matrix of the form: The matrices
can be related to the covariance matrices of
via the following expressions: Conversely: The cross-covariance matrix between two complex random vectors
are called uncorrelated if Two complex random vectors
denote the cumulative distribution functions of
denotes their joint cumulative distribution function.
are called independent if A complex random vector
is called circularly symmetric if for every deterministic
is called proper if the following three conditions are all satisfied:[1]: p. 293 Two complex random vectors
are called jointly proper is the composite random vector
The Cauchy-Schwarz inequality for complex random vectors is The characteristic function of a complex random vector