Let $x=\{x_1,x_2,...\}$ be a first-order autoregressive process, $AR(1)$. Since $x$ is an $AR(1)$ process, it is defined as follows:
$x_t=\mu+\phi x_{t-1}+\epsilon_t$
where $\mu$ and $\phi$ are constants, and $\epsilon_t$ is a $IID$ random variable distributed Normally with mean $0$ and variance $\sigma^2$.
Now consider another process, $y=\{y_1,y_2,...\}$ defined as follows:
$y_t = \left\{ \begin{array}{lr} x_t+y_{t-1} & : mod(t,a)\neq 1\\ x_t & : mod(t,a)= 1\\ \end{array} \right.\\$
where $a$ is a constant. How to calculate the mean and variance of process $y$?