4.8 Verify Eq. (4.68) to show that the three types of conditioning on the past are equivalent for a Gaussian QAR model, E[Xt|(Xt−s, s ∈ )] = ∑

s∈

asXt−s, E[Xt|(Xt−s, s ∈ )] = ∑

s∈

asXt−s, E[Xt|(Xt−s, s ∈ )] = ∑

s∈

asXt−s.

Hint: For simplicity work in d = 2 dimensions. The Gaussian zero-mean QAR model (4.66) is a special case of a UAR with respect to the lexicographic half-space  in (4.52). The spectral density is given by f(????) ????2

(2????)d

|

|

|

|

|

|

1 −

∑

s∈

aseisT????

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|

|

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−2

, and the innovation term, ????t = Xt − ∑

s∈asXt−s, is uncorrelated with Xt−s for all s ∈ .

Set

′ = {s ∈ ℤd ∶ s ≠ 0 and the last nonzero component s[????] of s is positive}.

to be the lexicographic half-space of ℝ2 obtained by interchanging the roles of t[1] and t[2] in the definition of  in (4.52). Then f(????) also defines a UAR with respect to ′

.

The innovation term is still ????t, as defined above, and is thus also uncorrelated with Xt−s for alls ∈ ′

. After writing Xt = ????t + ∑

s∈asXt−s, the desired result follows from the following properties: (i)  ⊂  ⊂  =  ∪ ′

, (ii) ????t is uncorrelated (and hence independent under the Gaussian assumption)

with Xt−s, s ∈ , and (iii) ∑

s∈asXt−s is constant given (Xt−s, s ∈ ).