4.4 Consider a d-dimensional AR model for a stationary Gaussian random field {Xt} given by sK dsXts = ????t, t d, where {????t}

4.4 Consider a d-dimensional AR model for a stationary Gaussian random field

{Xt} given by

∑

s∈K dsXt−s = ????t, t ∈ ℤd, where {????t} is white noise, and K ⊂ ℤd is a finite set. This framework includes both SARs (Section 4.5) and UARs (Section 4.8). Define a new sequence {as} with terms as = ∑

h∈D dh dh−s, s ∈ D(2)

, where D(2) = {h − s ∶ h,s ∈ D}. Show that this random field can be rewritten as a CAR model (4.31) with coefficients

????s = −as∕a0, s ∈ D(2)

.

Hence, conclude that every AR model can be written as a CAR model.

Hint: Work with the spectral density in both cases.