Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 1 / a

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- Solution a

Solution

For the autoregressive process of order one, the formula at

t = 0

reads

r_{0} = r_{0}

, with an empty sum. If it holds at time

t

, substitution into the recursion gives

r_{t + 1} = β (β^{t} r_{0} + \sum_{s = 1}^{t} β^{t - s} ξ_{s}) + ξ_{t + 1} = β^{t + 1} r_{0} + \sum_{s = 1}^{t + 1} β^{t + 1 - s} ξ_{s} .

Thus mathematical induction proves

r_{t} = β^{t} r_{0} + s = 1 \sum t β^{t - s} ξ_{s} .

No restriction such as

∣ β ∣ < 1

is needed for this finite-time identity.

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