Let $y\_n$ be a wide sense stationary process (*wss*), i.e., where the mean is independent of $n$ and the correlation depends solely upon $|n\_1 - n\_2|$.


Let us define:



> 
> $u\_n = a\_1 y\_{n−1} + \ldots + a\_p y\_{n-p}$
> 
> 
> 


with $a\_k \in \mathbb{R}$ and $k = 1, \ldots p$.


Is $u\_n$ also a wide sense stationary process? Why or why not?



wide sense stationary process
$u\_n$ is also a wide stationary process. Let $\mu=\mathsf{E}[y\_n]$ and $B(k)=\mathsf{cov}(y\_{n+k},y\_n)$. Then
\begin{gather}
\mathsf{E}[u\_n]=\mathsf{E}\Bigl[\sum\_{i=1}^pa\_iy\_{n-i}\Bigr]=\Bigl(\sum\_{i=1}^pa\_i\Bigr)\mu\quad\text {(it doesn't depend $n$)},\\
\mathsf{cov}(u\_{n+k},u\_n)=\mathsf{cov}\Bigl(\sum\_{i=1}^pa\_iy\_{n+k-i},\sum\_{j=1}^pa\_j{y\_{n-j}}\Bigr)=\sum\_{i=1}^p\sum\_{j=1}^pa\_ia\_jB(k-i+j)\\
\text {(it doesn't depend $n$ also)}.
\end{gather}
Therefore, $u\_n$ is a wide stationary process.


