 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-38/1/iv/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

For

∣ ϕ ∣ < 1

, the causal representation is

X_{t} = \sum_{j \geq 0} ϕ^{j} ε_{t - j}

. Thus

γ_{k} = \frac{σ ^{2}}{1 - ϕ ^{2}} ϕ^{∣ k ∣} .

Alternatively the rational spectrum from (iii) is

σ^{2} / {π (1 - ϕ z) (1 - ϕ z^{- 1})}

∣ z ∣ = 1

. The absolutely convergent geometric sums give

k \in Z \sum ϕ^{∣ k ∣} z^{k} = 1 + \frac{ϕ z}{1 - ϕ z} + \frac{ϕ z ^{- 1}}{1 - ϕ z ^{- 1}} = \frac{1 - ϕ ^{2}}{( 1 - ϕ z ) ( 1 - ϕ z ^{- 1} )} .

Consequently

f_{X} (ω) = \frac{σ ^{2}}{π ( 1 - ϕ ^{2} )} k \in Z \sum ϕ^{∣ k ∣} e^{ikω} .

This is the Poisson-kernel expansion of an AR(1) spectrum, with the same one-sided normalization as (i).

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