  We investigate large deviations properties for centered stationary AR(1) and
MA(1) processes with independent Gaussian innovations, by giving the explicit
bivariate rate functions for the sequence of random vectors
$(\boldsymbol{S}_n)_{n \in \N} = \left(n^{-1}(\sum_{k=1}^n X_k, \sum_{k=1}^n
X_k^2)\right)_{n \in \N}$. In the AR(1) case, we also give the explicit rate
function for the bivariate random sequence $(\W_n)_{n \geq 2} =
\left(n^{-1}(\sum_{k=1}^n X_k^2, \sum_{k=2}^n X_k X_{k+1})\right)_{n \geq
  2}$. Via Contraction Principle, we provide explicit rate functions for the
sequences $(n^{-1} \sum_{k=1}^n X_k)_{n \in \N}$, $(n^{-1} \sum_{k=1}^n
X_k^2)_{n \geq 2}$ and $(n^{-1} \sum_{k=2}^n X_k X_{k+1})_{n \geq 2}$, as well.
In the AR(1) case, we present a new proof for an already known result on the
explicit deviation function for the Yule-Walker estimator.
