Asymptotic behaviour of Gaussian processes with integral representation

The aim of this work is the study of limit points of Gaussian processes with continuous paths defined by an integral representation. Precisely, we control the asymptotic behaviour of a(t)Xt, where a is a function which vanishes at infinity and X is a Gaussian process. At first, we study real-valued processes indexed by a discrete set of parameters: we give a simple expression of the upper limit at infinity. We provide examples for which our assumptions hold but where the covariance decreases too slowly in order to use results of the existing literature. Next, we specialize our study to stationary processes with a spectral density and compute the set of limit values of a(t)Xt, where Xt is an n-dimensional Gaussian process. Finally, we establish results for the Ornstein–Uhlenbeck diffusion.