1. Definition

2. Cumulative distribution function

3. Expectation

4. Covariance matrix and pseudo-covariance matrix
    Definitions  Properties  Covariance matrices of real and imaginary parts 

5. Cross-covariance matrix and pseudo-cross-covariance matrix
    Definitions  Uncorrelatedness 

6. Independence

7. Circular symmetry
    Definition  Properties 

8. Proper complex random vectors
    Definition  Properties 

9. Cauchy-Schwarz inequality

10. Characteristic function

11. See also

12. References

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 real random vector on where denotes the real part of and denotes the imaginary part of .^[1]{{rp|p. 292}}

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 expessions 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.^[1]{{rp|p. 293}}

Covariance matrix and pseudo-covariance matrix

Definitions

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.^[2]{{rp|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. Thus the covariance matrix is a Hermitian matrix.^[1]{{rp|p. 293}}

The pseudo-covariance matrix (also called relation matrix) is defined as follows. In contrast to the covariance matrix defined above Hermitian transposition gets replaced by transposition in the definition.

Properties

The covariance matrix is a hermitian matrix, i.e.^[1]{{rp|p. 293}}

.

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 matrices and can be related to the covariance matrices of and via the following expressions:

and conversely

Cross-covariance matrix and pseudo-cross-covariance matrix

Definitions

The cross-covariance matrix between two complex random vectors is defined as:

And the pseudo-cross-covariance matrix is defined as:

Uncorrelatedness

Two complex random vectors and are called uncorrelated if

.

Independence

Two complex random vectors and are called independent if

where and denote the cumulative distribution functions of and as defined in {{EquationNote|Eq.1}} and denotes their joint cumulative distribution function. Independence of and is often denoted by .

Written component-wise, and are called independent if

.

Circular symmetry

Definition

A complex random vector is called circularly symmetric if for every deterministic the distribution of equals the distribution of .^[3]{{rp|pp. 500–501}}

Properties

• The expectation of a circularly symmetric complex random vectors is either zero or it is not defined.^[3]{{rp|p. 500}}
• The pseudo-covariance matrix of a circularly symmetric complex random vectors is zero.^[3]{{rp|p. 584}}

Proper complex random vectors

Definition

A complex random vector is called proper if the following three conditions are all satisfied:^[1]{{rp|p. 293}}

• (zero mean)
• (all components have finite variance)
• 

Two complex random vectors are called jointly proper is the composite random vector is proper.

Properties

• A complex random vector is proper if, and only if, for all (deterministic) vectors the complex random variable is proper.^[1]{{rp|p. 293}}
• Linear transformations of proper complex random vectors are proper, i.e. if is a proper random vectors with components and is a deterministic matrix, then the complex random vector is also proper.^[1]{{rp|p. 295}}
• Every circularly symmetric complex random vector with finite variance of all its components is proper.^[1]{{rp|p. 295}}
• There are proper complex random vectors that are not circularly symmetric.^[1]{{rp|p. 504}}
• A real random vector is proper if and only if it is constant.
• Two jointly proper complex random vectors are uncorrelated if and only if their covariace matrix is zero, i.e. if .

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality for complex random vectors is

.

Characteristic function

The characteristic function of a complex random vector with components is a function defined by:^[1]{{rp|p. 295}}

See also

• Complex random variable

References

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