6.2 The quadratic form Q(W) in the Whittle log-likelihood for a CAR model is specified as a linear combination of terms involving the biased sample covariance function. Show that it can also be expressed in terms of the periodogram as in (6.45).

Hint: Recall or prove the one-dimensional result for k ∈ ℤ, ∫ 2????

0 exp(ik????)

d???? = 2???? if k = 0, and 0 for k ≠ 0. Similarly, prove the d-dimensional result for k ∈ ℤd,

∫(−????,????)d exp(ikT????) d???? =

{(2????)

d, k = 0, 0, k ≠ 0.

Then expand out the integral ∫(−????,????)d I(????)∕f(????) d???? term by term.?