Explicit bivariate rate functions for large deviations in AR(1) and MA(1) processes with Gaussian innovations

We investigate the large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate (S_n)_n∈N =⁽ functions n⁻¹(^∑n k=1^Xk,^∑for the sequence of two-dimensional random vectors n. Via the Contraction Principle, we provide k=1^X2k⁾⁾n∈N the explicit rate functions for the sample mean and the sample second moment. In the AR(1) case, we also give the explicit rate function for the sequence of two-dimensional (W_n)_n⩾2 =⁽n⁻¹(^∑n k=1^X2k^,∑n random vectors k=2^XkX_k−1)⁾, but we obtain an n⩾2 analytic rate function that gives different values for the upper and lower bounds, depending on the evaluated set and its intersection with the respective set of exposed points. A careful analysis of the properties of a certain family of Toeplitz matrices is necessary. The large deviations properties of three particular sequences of one-dimensional random variables will follow after we show how to apply a weaker version of the Contraction Principle for our setting, providing new proofs for two already known results on the explicit deviation function for the sample second moment and Yule-Walker estimators. We exhibit the properties of the large deviations of the first-order empirical autocovariance, its explicit deviation function and this is also a new result.