Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 101 / 2 / b / Solution

Every $X_n$ is integrable: induction gives $|X_n|\le a^n|x|+{\sum }_{j=0}^{n-1}{a}^j$. Since the innovation has expectation zero, the recursion impliesStarting from $\mathbb{E}{X}_0=x$, induction therefore givesThis holds for every $a>0$, including $a=1$. The independence of the signs also identifies this as an autoregressive process of order one, although zero mean of each sign already suffices for the displayed expectation recursion.