208
In general, we may express the innovations process
α
(
n
) as a linear combination of the
observation vectors
y
(1),
y
(2),...,
y
(
n
) as follows:
where
A
n
1,0
=
I
. The set of matrices {
A
n
1,
k
} is chosen to satisfy the following conditions
We may thus write
The block lower triangular transformation matrix is invertible since its determinant equals
one. Hence, we may go back and forth between the given set of observation vectors
and
the
corresponding
set
of
innovations
processes
without any loss of information.
10.2
First, we note that
Since the estimate
consists of a linear combination of the observation vectors
, and since
y
1
( )
y
2
( )
I
0
A
1 1
,
I
1
–
α
1
( )
α
2
( )
=
I
0
A
–
1 1
,
I
α
1
( )
α
2
( )
=
α
n
( )
y
n
( )
A
1 1
,
y
n
1
–
(
)
…
A
n
1
n
1
,
y
1
( )
+
+
+
=
A
n
1
k
,
,
y
n

k
+1
(
),
n
k
=1
n
∑
1 2
…
,
,
,
=
=
E
α
n
+1
(
)α
H
n
( )
[
]
0
=
n
1 2
…
,
,
,
=
α
1
( )
α
2
( )
α
n
( )
I
0
…
0
A
1 1
,
I
…
0
A
n
1
n
1
,
,
A
n
1
n
2
,
,
…
I
y
1
( )
y
2
( )
y
n
( )
=
. . .
. . .
. . .
. . .
. . .
y
1
( )
y
2
( ) …
y
n
( )
,
,
,
{
}
α
1
( ) α
2
( ) … α
n
( )
,
,
,
{
}
E
ε
n n
1
,
(
)
v
1
H
n
( )
[
]
E
x
n
( )
v
1
H
n
( )
[
]
E
x
ˆ
n
y
n
1
(
)
v
1
H
n
( )
[
]
–
=
x
ˆ
n
y
n
1
–
(
)
y
1
( ) …
y
n
1
(
)
,
,