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.
Categories: math.PR math.DS math.ST stat.TH