Continuous Breuer-Major theorem: tightness and nonstationarity (Q2184814)

The paper deals with the celebrated Breuer-Major theorem (in its continuous version) which asserts that under some additional conditions the finite-dimensional distributions of integral functional of a zero-mean Gaussian stationary process tend to those of \(\sigma W\), where \(W\) is a standard Brownian motion and \(\sigma\) is some explicit constant. The goal of this paper is twofold. First, the authors investigate the tightness in the Breuer-Major theorem. Their condition of tightness is very simple, contrary to previous ones, as it only requires that \(|f|^p\) must be integrable with respect to the standard Gaussian measure for some \(p\) strictly bigger than 2. This result is obtained by means of the Malliavin calculus, in particular with the help of Meyer inequalities. Second, and motivated by a problem of geometrical nature, the authors extend the continuous Breuer-Major theorem to the notoriously difficult case of selfsimilar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Brownian motion concludes the paper.