<s>
In	O
probability	O
,	O
and	O
statistics	O
,	O
a	O
multivariate	B-General_Concept
random	I-General_Concept
variable	I-General_Concept
or	O
random	B-General_Concept
vector	I-General_Concept
is	O
a	O
list	O
or	O
vector	O
of	O
mathematical	O
variables	O
each	O
of	O
whose	O
value	O
is	O
unknown	O
,	O
either	O
because	O
the	O
value	O
has	O
not	O
yet	O
occurred	O
or	O
because	O
there	O
is	O
imperfect	O
knowledge	O
of	O
its	O
value	O
.	O
</s>
<s>
The	O
individual	O
variables	O
in	O
a	O
random	B-General_Concept
vector	I-General_Concept
are	O
grouped	O
together	O
because	O
they	O
are	O
all	O
part	O
of	O
a	O
single	O
mathematical	O
system	O
—	O
often	O
they	O
represent	O
different	O
properties	O
of	O
an	O
individual	O
statistical	O
unit	O
.	O
</s>
<s>
For	O
example	O
,	O
while	O
a	O
given	O
person	O
has	O
a	O
specific	O
age	O
,	O
height	O
and	O
weight	O
,	O
the	O
representation	O
of	O
these	O
features	O
of	O
an	O
unspecified	O
person	O
from	O
within	O
a	O
group	O
would	O
be	O
a	O
random	B-General_Concept
vector	I-General_Concept
.	O
</s>
<s>
Normally	O
each	O
element	O
of	O
a	O
random	B-General_Concept
vector	I-General_Concept
is	O
a	O
real	O
number	O
.	O
</s>
<s>
Random	B-General_Concept
vectors	I-General_Concept
are	O
often	O
used	O
as	O
the	O
underlying	O
implementation	O
of	O
various	O
types	O
of	O
aggregate	O
random	O
variables	O
,	O
e.g.	O
</s>
<s>
a	O
random	O
matrix	B-Architecture
,	O
random	O
tree	O
,	O
random	O
sequence	O
,	O
stochastic	O
process	O
,	O
etc	O
.	O
</s>
<s>
More	O
formally	O
,	O
a	O
multivariate	B-General_Concept
random	I-General_Concept
variable	I-General_Concept
is	O
a	O
column	O
vector	O
(	O
or	O
its	O
transpose	O
,	O
which	O
is	O
a	O
row	O
vector	O
)	O
whose	O
components	O
are	O
scalar-valued	O
random	O
variables	O
on	O
the	O
same	O
probability	O
space	O
as	O
each	O
other	O
,	O
,	O
where	O
is	O
the	O
sample	O
space	O
,	O
is	O
the	O
sigma-algebra	O
(	O
the	O
collection	O
of	O
all	O
events	O
)	O
,	O
and	O
is	O
the	O
probability	O
measure	O
(	O
a	O
function	O
returning	O
each	O
event	O
's	O
probability	O
)	O
.	O
</s>
<s>
Every	O
random	B-General_Concept
vector	I-General_Concept
gives	O
rise	O
to	O
a	O
probability	O
measure	O
on	O
with	O
the	O
Borel	O
algebra	O
as	O
the	O
underlying	O
sigma-algebra	O
.	O
</s>
<s>
This	O
measure	O
is	O
also	O
known	O
as	O
the	O
joint	O
probability	O
distribution	O
,	O
the	O
joint	O
distribution	O
,	O
or	O
the	O
multivariate	O
distribution	O
of	O
the	O
random	B-General_Concept
vector	I-General_Concept
.	O
</s>
<s>
Random	B-General_Concept
vectors	I-General_Concept
can	O
be	O
subjected	O
to	O
the	O
same	O
kinds	O
of	O
algebraic	O
operations	O
as	O
can	O
non-random	O
vectors	O
:	O
addition	O
,	O
subtraction	O
,	O
multiplication	O
by	O
a	O
scalar	O
,	O
and	O
the	O
taking	O
of	O
inner	O
products	O
.	O
</s>
<s>
Similarly	O
,	O
a	O
new	O
random	B-General_Concept
vector	I-General_Concept
can	O
be	O
defined	O
by	O
applying	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
to	O
a	O
random	B-General_Concept
vector	I-General_Concept
:	O
</s>
<s>
,	O
where	O
is	O
an	O
matrix	B-Architecture
and	O
is	O
an	O
column	O
vector	O
.	O
</s>
<s>
More	O
generally	O
we	O
can	O
study	O
invertible	O
mappings	O
of	O
random	B-General_Concept
vectors	I-General_Concept
.	O
</s>
<s>
Assume	O
that	O
the	O
real	O
random	B-General_Concept
vector	I-General_Concept
has	O
a	O
probability	O
density	O
function	O
and	O
satisfies	O
.	O
</s>
<s>
The	O
expected	O
value	O
or	O
mean	O
of	O
a	O
random	B-General_Concept
vector	I-General_Concept
is	O
a	O
fixed	O
vector	O
whose	O
elements	O
are	O
the	O
expected	O
values	O
of	O
the	O
respective	O
random	O
variables	O
.	O
</s>
<s>
The	O
covariance	O
matrix	B-Architecture
(	O
also	O
called	O
second	O
central	O
moment	O
or	O
variance-covariance	O
matrix	B-Architecture
)	O
of	O
an	O
random	B-General_Concept
vector	I-General_Concept
is	O
an	O
matrix	B-Architecture
whose	O
(	O
i	O
,	O
j	O
)	O
th	O
element	O
is	O
the	O
covariance	O
between	O
the	O
i	O
th	O
and	O
the	O
j	O
th	O
random	O
variables	O
.	O
</s>
<s>
The	O
covariance	O
matrix	B-Architecture
is	O
the	O
expected	O
value	O
,	O
element	O
by	O
element	O
,	O
of	O
the	O
matrix	B-Architecture
computed	O
as	O
,	O
where	O
the	O
superscript	O
T	O
refers	O
to	O
the	O
transpose	O
of	O
the	O
indicated	O
vector	O
:	O
</s>
<s>
where	O
again	O
the	O
matrix	B-Architecture
expectation	O
is	O
taken	O
element-by-element	O
in	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
covariance	O
matrix	B-Architecture
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
i.e.	O
</s>
<s>
The	O
covariance	O
matrix	B-Architecture
is	O
a	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
,	O
i.e.	O
</s>
<s>
The	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
is	O
simply	O
the	O
transpose	O
of	O
the	O
matrix	B-Architecture
,	O
i.e.	O
</s>
<s>
They	O
are	O
uncorrelated	O
if	O
and	O
only	O
if	O
their	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
is	O
zero	O
.	O
</s>
<s>
The	O
correlation	B-Algorithm
matrix	I-Algorithm
(	O
also	O
called	O
second	O
moment	O
)	O
of	O
an	O
random	B-General_Concept
vector	I-General_Concept
is	O
an	O
matrix	B-Architecture
whose	O
(	O
i	O
,	O
j	O
)	O
th	O
element	O
is	O
the	O
correlation	O
between	O
the	O
i	O
th	O
and	O
the	O
j	O
th	O
random	O
variables	O
.	O
</s>
<s>
The	O
correlation	B-Algorithm
matrix	I-Algorithm
is	O
the	O
expected	O
value	O
,	O
element	O
by	O
element	O
,	O
of	O
the	O
matrix	B-Architecture
computed	O
as	O
,	O
where	O
the	O
superscript	O
T	O
refers	O
to	O
the	O
transpose	O
of	O
the	O
indicated	O
vector	O
:	O
</s>
<s>
Similarly	O
for	O
the	O
cross-correlation	O
matrix	O
and	O
the	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
The	O
characteristic	O
function	O
of	O
a	O
random	B-General_Concept
vector	I-General_Concept
with	O
components	O
is	O
a	O
function	O
that	O
maps	O
every	O
vector	O
to	O
a	O
complex	O
number	O
.	O
</s>
<s>
One	O
can	O
take	O
the	O
expectation	O
of	O
a	O
quadratic	O
form	O
in	O
the	O
random	B-General_Concept
vector	I-General_Concept
as	O
follows	O
:	O
</s>
<s>
where	O
is	O
the	O
covariance	O
matrix	B-Architecture
of	O
and	O
refers	O
to	O
the	O
trace	O
of	O
a	O
matrix	B-Architecture
—	O
that	O
is	O
,	O
to	O
the	O
sum	O
of	O
the	O
elements	O
on	O
its	O
main	O
diagonal	O
(	O
from	O
upper	O
left	O
to	O
lower	O
right	O
)	O
.	O
</s>
<s>
Proof	O
:	O
Let	O
be	O
an	O
random	B-General_Concept
vector	I-General_Concept
with	O
and	O
and	O
let	O
be	O
an	O
non-stochastic	O
matrix	B-Architecture
.	O
</s>
<s>
One	O
can	O
take	O
the	O
expectation	O
of	O
the	O
product	O
of	O
two	O
different	O
quadratic	O
forms	O
in	O
a	O
zero-mean	O
Gaussian	O
random	B-General_Concept
vector	I-General_Concept
as	O
follows	O
:	O
</s>
<s>
where	O
again	O
is	O
the	O
covariance	O
matrix	B-Architecture
of	O
.	O
</s>
<s>
Here	O
the	O
random	B-General_Concept
vector	I-General_Concept
is	O
the	O
vector	O
of	O
random	O
returns	O
on	O
the	O
individual	O
assets	O
,	O
and	O
the	O
portfolio	O
return	O
p	O
(	O
a	O
random	O
scalar	O
)	O
is	O
the	O
inner	O
product	O
of	O
the	O
vector	O
of	O
random	O
returns	O
with	O
a	O
vector	O
w	O
of	O
portfolio	O
weights	O
—	O
the	O
fractions	O
of	O
the	O
portfolio	O
placed	O
in	O
the	O
respective	O
assets	O
.	O
</s>
<s>
Since	O
p	O
=	O
wT	O
,	O
the	O
expected	O
value	O
of	O
the	O
portfolio	O
return	O
is	O
wTE( )	O
and	O
the	O
variance	O
of	O
the	O
portfolio	O
return	O
can	O
be	O
shown	O
to	O
be	O
wTCw	O
,	O
where	O
C	O
is	O
the	O
covariance	O
matrix	B-Architecture
of	O
.	O
</s>
<s>
In	O
linear	B-General_Concept
regression	I-General_Concept
theory	O
,	O
we	O
have	O
data	O
on	O
n	O
observations	O
on	O
a	O
dependent	O
variable	O
y	O
and	O
n	O
observations	O
on	O
each	O
of	O
k	O
independent	O
variables	O
xj	O
.	O
</s>
<s>
The	O
observations	O
on	O
the	O
dependent	O
variable	O
are	O
stacked	O
into	O
a	O
column	O
vector	O
y	O
;	O
the	O
observations	O
on	O
each	O
independent	O
variable	O
are	O
also	O
stacked	O
into	O
column	O
vectors	O
,	O
and	O
these	O
latter	O
column	O
vectors	O
are	O
combined	O
into	O
a	O
design	B-Algorithm
matrix	I-Algorithm
X	O
(	O
not	O
denoting	O
a	O
random	B-General_Concept
vector	I-General_Concept
in	O
this	O
context	O
)	O
of	O
observations	O
on	O
the	O
independent	O
variables	O
.	O
</s>
<s>
where	O
β	O
is	O
a	O
postulated	O
fixed	O
but	O
unknown	O
vector	O
of	O
k	O
response	O
coefficients	O
,	O
and	O
e	O
is	O
an	O
unknown	O
random	B-General_Concept
vector	I-General_Concept
reflecting	O
random	O
influences	O
on	O
the	O
dependent	O
variable	O
.	O
</s>
<s>
Then	O
the	O
statistician	O
must	O
analyze	O
the	O
properties	O
of	O
and	O
,	O
which	O
are	O
viewed	O
as	O
random	B-General_Concept
vectors	I-General_Concept
since	O
a	O
randomly	O
different	O
selection	O
of	O
n	O
cases	O
to	O
observe	O
would	O
have	O
resulted	O
in	O
different	O
values	O
for	O
them	O
.	O
</s>
<s>
The	O
evolution	O
of	O
a	O
k×1	O
random	B-General_Concept
vector	I-General_Concept
through	O
time	O
can	O
be	O
modelled	O
as	O
a	O
vector	O
autoregression	O
(	O
VAR	O
)	O
as	O
follows	O
:	O
</s>
<s>
where	O
the	O
i-periods-back	O
vector	O
observation	O
is	O
called	O
the	O
i-th	O
lag	O
of	O
,	O
c	O
is	O
a	O
k×1	O
vector	O
of	O
constants	O
(	O
intercepts	B-Algorithm
)	O
,	O
Ai	O
is	O
a	O
time-invariant	O
k×k	O
matrix	B-Architecture
and	O
is	O
a	O
k×1	O
random	B-General_Concept
vector	I-General_Concept
of	O
error	O
terms	O
.	O
</s>
