I have an $\operatorname{ARMA}(p,q)$ process $X_n$, defined say by $$ \varphi(L)X_n=\vartheta(L)\varepsilon_n. $$ with i.i.d. Gaussian innovation terms $\varepsilon_n$.
I can observe this process only at a reduced frequency, once every $k$ time steps. That means that I observe the process $Y_m=X_{km}$ for some integer $k>1$.
My Questions
Is the process $Y_m$ also representable as an ARMA process? Or does it belong to some other larger class of stochastic processes?
If $Y_m$ is effectively an ARMA process, is there an explicit procedure to compute its coefficients?
Special cases
The question has an affirmative answer in some special cases:
If $X_n$ is an $\operatorname{AR}(1)$ process coefficient $\varphi$, then $Y_m$ is also an $\operatorname{AR}(1)$ process with coefficient $\varphi^k$ and innovation terms $\sum_{i=1}^{k}\varphi^{k-i}\varepsilon_{km-i+1}$.
If $X_t$ is a continuous Ornstein-Uhlenbeck process and $Y_m$ is a sampling of $X_t$ at regular intervals, then $Y_m$ is an $\operatorname{AR}(1)$ process.
If $X_n$ is an $\operatorname{MA}(q)$ process with $q\leq k$ then $Y_m$ is a white noise process.
PS. If it may be useful, the process $X_n$ can be assumed to be weakly or strongly stationary.