# Complex random vector

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 $Z_{1},\ldots ,Z_{n}$ are complex-valued random variables, then the n-tuple $\left(Z_{1},\ldots ,Z_{n}\right)$ 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 $\mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{T}$ on the probability space $(\Omega ,{\mathcal {F}},P)$ is a function $\mathbf {Z} \colon \Omega \rightarrow \mathbb {C} ^{n}$ such that the vector $(\Re {(Z_{1})},\Im {(Z_{1})},\ldots ,\Re {(Z_{n})},\Im {(Z_{n})})^{T}$ is a real random vector on $(\Omega ,{\mathcal {F}},P)$ where $\Re {(z)}$ denotes the real part of $z$ and $\Im {(z)}$ denotes the imaginary part of $z$.$^{:p.292}$

## Cumulative distribution function

The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form $P(Z\leq 1+3i)$ make no sense. However expressions of the form $P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)$ make sense. Therefore, the cumulative distribution function $F_{\mathbf {Z} }:\mathbb {C} ^{n}\mapsto [0,1]$ of a random vector $\mathbf {Z} =(Z_{1},...,Z_{n})^{T}$ is defined as
| $F_{\mathbf {Z} }(\mathbf {z} )=\operatorname {P} (\Re {(Z_{1})}\leq \Re {(z_{1})},\Im {(Z_{1})}\leq \Im {(z_{1})},\ldots ,\Re {(Z_{n})}\leq \Re {(z_{n})},\Im {(Z_{n})}\leq \Im {(z_{n})})$ | (Eq.1) |
where $\mathbf {z} =(z_{1},...,z_{n})^{T}$.

## Expectation

As in the real case the expectation (also called expected value) of a complex random vector is taken component-wise.$^{:p.293}$
| $\operatorname {E} [\mathbf {Z} ]=(\operatorname {E} [Z_{1}],\ldots ,\operatorname {E} [Z_{n}])^{T}$ | (Eq.2) |

## Covariance matrix and pseudo-covariance matrix

The covariance matrix (also called second central moment) $\operatorname {K} _{\mathbf {Z} \mathbf {Z} }$ contains the covariances between all pairs of components. The covariance matrix of an $n\times 1$ random vector is an $n\times n$ matrix whose $(i,j)$th element is the covariance between the i th and the j th random variables.$^{:p.372}$ 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.$^{:p.293}$
| ${\begin{aligned}&\operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} [\mathbf {Z} ,\mathbf {Z} ]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])}^{H}]=\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{H}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ^{H}]\\[12pt]\end{aligned}}$ | (Eq.3) |
    $\operatorname {K} _{\mathbf {Z} \mathbf {Z} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\end{bmatrix}}$
The pseudo-covariance matrix (also called relation matrix) is defined replacing Hermitian transposition by transposition in the definition above.
| $\operatorname {J} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} [\mathbf {Z} ,{\overline {\mathbf {Z} }}]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])}^{T}]=\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ^{T}]$ | (Eq.4) |
    $\operatorname {J} _{\mathbf {Z} \mathbf {Z} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{n}-\operatorname {E} [Z_{n}])]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{n}-\operatorname {E} [Z_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{n}-\operatorname {E} [Z_{n}])]\end{bmatrix}}$

### Properties

The covariance matrix is a hermitian matrix, i.e.$^{:p.293}$
    $\operatorname {K} _{\mathbf {Z} \mathbf {Z} }^{H}=\operatorname {K} _{\mathbf {Z} \mathbf {Z} }$.
The pseudo-covariance matrix is a symmetric matrix, i.e.
    $\operatorname {J} _{\mathbf {Z} \mathbf {Z} }^{T}=\operatorname {J} _{\mathbf {Z} \mathbf {Z} }$.
The covariance matrix is a positive semidefinite matrix, i.e.
    $\mathbf {a} ^{H}\operatorname {K} _{\mathbf {Z} \mathbf {Z} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {C} ^{n}$.

### Covariance matrices of real and imaginary parts

By decomposing the random vector $\mathbf {Z}$ into its real part $\mathbf {X} =\Re {(\mathbf {Z} )}$ and imaginary part $\mathbf {Y} =\Im {(\mathbf {Z} )}$ (i.e. $\mathbf {Z} =\mathbf {X} +i\mathbf {Y}$), the pair $(\mathbf {X} ,\mathbf {Y} )$ has a covariance matrix of the form:
    ${\begin{bmatrix}\operatorname {K} _{\mathbf {X} \mathbf {X} }&\operatorname {K} _{\mathbf {X} \mathbf {Y} }\\\operatorname {K} _{\mathbf {Y} \mathbf {X} }&\operatorname {K} _{\mathbf {Y} \mathbf {Y} }\end{bmatrix}}$
The matrices $\operatorname {K} _{\mathbf {Z} \mathbf {Z} }$ and $\operatorname {J} _{\mathbf {Z} \mathbf {Z} }$ can be related to the covariance matrices of $\mathbf {X}$ and $\mathbf {Y}$ via the following expressions:
    ${\begin{aligned}&\operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{\mathbf {Z} \mathbf {Z} }+\operatorname {J} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {Y} \mathbf {Y} }=\operatorname {E} [(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{\mathbf {Z} \mathbf {Z} }-\operatorname {J} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {Y} \mathbf {X} }=\operatorname {E} [(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{\mathbf {Z} \mathbf {Z} }+\operatorname {K} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{\mathbf {Z} \mathbf {Z} }-\operatorname {K} _{\mathbf {Z} \mathbf {Z} })\\\end{aligned}}$
Conversely:
    ${\begin{aligned}&\operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {K} _{\mathbf {X} \mathbf {X} }+\operatorname {K} _{\mathbf {Y} \mathbf {Y} }+i(\operatorname {K} _{\mathbf {Y} \mathbf {X} }-\operatorname {K} _{\mathbf {X} \mathbf {Y} })\\&\operatorname {J} _{\mathbf {Z} \mathbf {Z} }=\operatorname {K} _{\mathbf {X} \mathbf {X} }-\operatorname {K} _{\mathbf {Y} \mathbf {Y} }+i(\operatorname {K} _{\mathbf {Y} \mathbf {X} }+\operatorname {K} _{\mathbf {X} \mathbf {Y} })\end{aligned}}$

## Cross-covariance matrix and pseudo-cross-covariance matrix

The cross-covariance matrix between two complex random vectors $\mathbf {Z} ,\mathbf {W}$ is defined as:
| $\operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} [\mathbf {Z} ,\mathbf {W} ]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{H}]=\operatorname {E} [\mathbf {Z} \mathbf {W} ^{H}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ^{H}]$ | (Eq.5) |
    $\operatorname {K} _{\mathbf {Z} \mathbf {W} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\end{bmatrix}}$
And the pseudo-cross-covariance matrix is defined as:
| $\operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} [\mathbf {Z} ,{\overline {\mathbf {W} }}]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{T}]=\operatorname {E} [\mathbf {Z} \mathbf {W} ^{T}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ^{T}]$ | (Eq.6) |
    $\operatorname {J} _{\mathbf {Z} \mathbf {W} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{n}-\operatorname {E} [W_{n}])]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{n}-\operatorname {E} [W_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{n}-\operatorname {E} [W_{n}])]\end{bmatrix}}$
Two complex random vectors $\mathbf {Z}$ and $\mathbf {W}$ are called uncorrelated if
    $\operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0$.

## Independence

Two complex random vectors $\mathbf {Z} =(Z_{1},...,Z_{m})^{T}$ and $\mathbf {W} =(W_{1},...,W_{n})^{T}$ are called independent if
| $F_{\mathbf {Z,W} }(\mathbf {z,w} )=F_{\mathbf {Z} }(\mathbf {z} )\cdot F_{\mathbf {W} }(\mathbf {w} )\quad {\text{for all }}\mathbf {z} ,\mathbf {w}$ | (Eq.7) |
where $F_{\mathbf {Z} }(\mathbf {z} )$ and $F_{\mathbf {W} }(\mathbf {w} )$ denote the cumulative distribution functions of $\mathbf {Z}$ and $\mathbf {W}$ as defined in Eq.1 and $F_{\mathbf {Z,W} }(\mathbf {z,w} )$ denotes their joint cumulative distribution function. Independence of $\mathbf {Z}$ and $\mathbf {W}$ is often denoted by $\mathbf {Z} \perp \!\!\!\perp \mathbf {W}$. Written component-wise, $\mathbf {Z}$ and $\mathbf {W}$ are called independent if
    $F_{Z_{1},\ldots ,Z_{m},W_{1},\ldots ,W_{n}}(z_{1},\ldots ,z_{m},w_{1},\ldots ,w_{n})=F_{Z_{1},\ldots ,Z_{m}}(z_{1},\ldots ,z_{m})\cdot F_{W_{1},\ldots ,W_{n}}(w_{1},\ldots ,w_{n})\quad {\text{for all }}z_{1},\ldots ,z_{m},w_{1},\ldots ,w_{n}$.

## Circular symmetry

A complex random vector $\mathbf {Z}$ is called circularly symmetric if for every deterministic $\varphi \in [-\pi ,\pi )$ the distribution of $e^{\mathrm {i} \varphi }\mathbf {Z}$ equals the distribution of $\mathbf {Z}$.$^{:pp.500–501}$

### Properties

* The expectation of a circularly symmetric complex random vector is either zero or it is not defined.$^{:p.500}$
* The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.$^{:p.584}$

## Proper complex random vectors

A complex random vector $\mathbf {Z}$ is called proper if the following three conditions are all satisfied:$^{:p.293}$
* $\operatorname {E} [\mathbf {Z} ]=0$ (zero mean)
* $\operatorname {var} [Z_{1}]<\infty ,\ldots ,\operatorname {var} [Z_{n}]<\infty$ (all components have finite variance)
* $\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]=0$
Two complex random vectors $\mathbf {Z} ,\mathbf {W}$ are called jointly proper is the composite random vector $(Z_{1},Z_{2},\ldots ,Z_{m},W_{1},W_{2},\ldots ,W_{n})^{T}$ is proper.

### Properties

* A complex random vector $\mathbf {Z}$ is proper if, and only if, for all (deterministic) vectors $\mathbf {c} \in \mathbb {C} ^{n}$ the complex random variable $\mathbf {c} ^{T}\mathbf {Z}$ is proper.$^{:p.293}$
* Linear transformations of proper complex random vectors are proper, i.e. if $\mathbf {Z}$ is a proper random vectors with $n$ components and $A$ is a deterministic $m\times n$ matrix, then the complex random vector $A\mathbf {Z}$ is also proper.$^{:p.295}$
* Every circularly symmetric complex random vector with finite variance of all its components is proper.$^{:p.295}$
* There are proper complex random vectors that are not circularly symmetric.$^{:p.504}$
* 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 covariance matrix is zero, i.e. if $\operatorname {K} _{\mathbf {Z} \mathbf {W} }=0$.

## Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality for complex random vectors is
    $\left|\operatorname {E} [\mathbf {Z} ^{H}\mathbf {W} ]\right|^{2}\leq \operatorname {E} [\mathbf {Z} ^{H}\mathbf {Z} ]\operatorname {E} [|\mathbf {W} ^{H}\mathbf {W} |]$.

## Characteristic function

The characteristic function of a complex random vector $\mathbf {Z}$ with $n$ components is a function $\mathbb {C} ^{n}\to \mathbb {C}$ defined by:$^{:p.295}$
    $\varphi _{\mathbf {Z} }(\mathbf {\omega } )=\operatorname {E} \left[e^{i\Re {(\mathbf {\omega } ^{H}\mathbf {Z} )}}\right]=\operatorname {E} \left[e^{i(\Re {(\omega _{1})}\Re {(Z_{1})}+\Im {(\omega _{1})}\Im {(Z_{1})}+\cdots +\Re {(\omega _{n})}\Re {(Z_{n})}+\Im {(\omega _{n})}\Im {(Z_{n})})}\right]$