Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 2 / d / i

Table of contents
- Solution i

Solution

The autoregressive process of order one is causal exactly when

∣ ϕ ∣ < 1,

because then

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

converges in mean square. Its autocovariance is

γ_{X} (h) = \frac{σ ^{2}}{1 - ϕ ^{2}} ϕ^{∣ h ∣}, h \in Z .

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