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8.1 State-Space Representations 261
Remark 3. It follows from the observation equation (8.1.1) and the state equation
(8.1.2) that Xt and Yt have the functional forms, for t 2, 3,...,
and
Xt Ft−1Xt−1 + Vt−1
Ft−1(Ft−2Xt−2 + Vt−2) + Vt−1
.
(Ft−1 ···F1)X1 + (Ft−1 ···F2)V1 +···+Ft−1Vt−2 + Vt−1
ft(X1, V1,...,Vt−1) (8.1.3)
Yt gt(X1, V1,...,Vt−1, Wt). (8.1.4)
Remark 4. From Remark 3 and the assumptions on the noise terms, it is clear that
0, 1 ≤ s ≤ t,
and
E VtX ′
s 0, E VtY ′
s
E WtX ′
s
0, 1 ≤ s ≤ t, E(WtY ′
s ) 0, 1 ≤ s