G\"artner-Ellis condition for squared asymptotically stationary Gaussian processes

TL;DR

This paper establishes the G"artner-Ellis condition for squared asymptotically stationary Gaussian processes, revealing the limit distribution as a convolution of Gamma and exponential-Poisson distributions, with implications for ergodicity.

Contribution

It extends the G"artner-Ellis condition to squared Gaussian processes and provides a probabilistic interpretation via Wiener-Hopf factorization.

Findings

Limit distribution is the Laplace transform of a convolution of Gamma distributions.

The same limit applies conditionally given any initial point.

The approach offers a new perspective on weak multiplicative ergodicity.

Abstract

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.