The Gärtner-Ellis condition for the square of an asymptotically stationary Gaussian process is established. The same limit holds for the conditional distri-bution given any fixed initial point, which entails weak multiplicative ergodicity. The limit is shown to be the Laplace transform of a convolution of Gamma distributions with Poisson compound of exponentials. A proof based on Wiener-Hopf factorization induces a probabilistic interpretation of the limit in terms of a regression problem.