 Solution

For the spectral questions take the unique weakly stationary process solving the equation, as is customary for a stable autoregressive moving-average model. The equation alone also admits nonstationary solutions differing by $C{4}^{-t}$, for which a spectral density of a stationary process need not exist. Let $B$ be the backshift operator. Since the autoregressive root is $4$, the stationary solution is causal andUsing the convention $\gamma (h)={\int }_{-\pi }^{\pi }{e}^{ih\lambda }f(\lambda )d\lambda$, the spectral density of a stationary process isThe stationary mean is zero, since $\mu -\mu /4=0$.