We establish the Gärtner–Ellis condition for the square of an asymptotically stationary Gaussian process. The same limit holds for the conditional distribution 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 the Wiener–Hopf factorization induces a probabilistic interpretation of the limit in terms of a regression problem.