The cross-covariance matrix between two random vectors
and
is a matrix containing the
covariances between all
the possible couples of random variables formed by one entry of
and one entry of
.
This is a formal definition.
Definition
Let
be a
random vector and
be a
random vector. The cross-covariance matrix between
and
is a
matrix, denoted by
and defined as
follows:![[eq2]](/images/cross-covariance-matrix__13.png){kind=small}
Note that in the formula above
is a column vector and
is a row vector.
Define two random vectors
and
as
follows:
The cross-covariance matrix between
and
is![[eq6]](/images/cross-covariance-matrix__21.png)
When
,
then the cross-covariance matrix coincides with the
covariance
matrix of
:![[eq7]](/images/cross-covariance-matrix__24.png){kind=small}
Let
be a sequence of random vectors, where
is a time-index.
Let
be a time lag. Then, the cross-covariance
matrixis
called autocovariance matrix.
Note that, in general, the cross-covariance is not symmetric.
In
fact,and,
in
general,
For example, if
is
and
is
,
is
and
is
.
We have just demonstrated that
Other useful properties that can be easily derived from the definition (exercise!) are:
formula in terms of
cross-moments:
additivity:![[eq16]](/images/cross-covariance-matrix__41.png)
homogeneity:where
is a constant scalar;
formula for linear
transformations:where
and
are constant vectors and
and
are constant conformable matrices.
The cross-covariance matrix is often used in time-series analysis and in the theory of stochastic processes.
For example, it is used to define the concept of covariance stationarity for random vectors.
A sequence of random
vectors
is said to be covariance
stationary if and only
if![[eq20]](/images/cross-covariance-matrix__50.png)
where
and
are integers.
Property (1) means that all the random vectors belonging to the sequence
must have the same mean.
Property (2) means that the cross-covariance between a term of the sequence
()
and the term that is located
positions before it
()
must always be the same, irrespective of how
has been chosen. In other words,
depends only on
and not on
.
If
is a covariance stationary sequence of vectors, we can use the following
important formula for the covariance matrix of a
sum:![[eq24]](/images/cross-covariance-matrix__62.png)
This is demonstrated as
follows:![[eq25]](/images/cross-covariance-matrix__63.png)
where:
in step
we have used the fact that the cross-covariance of a vector with itself is
equal to its covariance matrix; in step
we have used the additivity property; in step
we have used the stationarity of the sequence; in step
we have re-grouped the summands.
Consider a sequence of covariance stationary random vectors
.
The sample mean of the first
terms of the sequence
is
Under some regularity conditions, the sample mean is asymptotically
normal, with
mean
and asymptotic covariance
matrix![[eq29]](/images/cross-covariance-matrix__72.png)
where
the cross-covariance matrices
are defined as in equation (2) above.
In other words, if a multivariate
central limit theorem
for dependent sequences applies,
thenwhere
denotes convergence
in distribution and
denotes a
multivariate
normal distribution with zero mean and long-run covariance matrix equal to
.
Let
be a sequence of random vectors.
When the
cross-covariance![[eq34]](/images/cross-covariance-matrix__79.png){kind=small}
is
viewed as a function of the two indices
and
,
then it is called cross-covariance function.
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Please cite as:
Taboga, Marco (2021). "Cross-covariance matrix", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/cross-covariance-matrix.
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