Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 3 / c / Solution

Iterating the autoregressive process of order one givesFor $q_r={\sum }_{j=r}^t{\theta }_j{a}^{j-r}$, regrouping the sum of the exponents givesThe independent innovations factorize the moment-generating function. ConsequentlyFinite sums avoid a division by $a-1$, so these formulas include $a=0$ and $a=1$; zeroth powers in these sums are $1$.