  The G\"artner-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.
