Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 36 / 2 / b / Solution

The condition for a causal linear-filter solution is $|\varphi |<1$. Its mean-square expansion is $X_t={\sum }_{j\ge 0}{\varphi }^j{Z}_{t-j}$. Summing the matching white noise terms givesThe assumed orthogonality of every $W_s$ to every $Z_t$ extends to every $X_t$ by L2 convergence of that expansion. Hence the added-noise process has mean zero andThis depends only on lag, proving weak stationarity. Its autocorrelation has the same geometric tail as the latent autoregression, but its positive-lag correlations are reduced by the additional variance at lag zero. This is the autocovariance of an AR(1) process observed with white noise. Strict stationarity or Gaussianity is not implied by white noise covariance assumptions alone.