On weak convergence in dynamical systems to self-similar processes with spectral representation

Abstract:

Let $(X,\mu ,T)$ be an aperiodic dynamical system. Set ${S_m}f = f + \cdots + f \circ {T^{m - 1}}$, where $f$ is a measurable function on $X$. Let $Y(t)$ be one of a class of self-similar process with a “nice” spectral representation, for instance, either a fractional Brownian motion, a Hermite process, or a harmonizable fractional stable motion. We show that there is an $f$ on $X$, and constants ${A_m} \to + \infty$ so that \[ A_m^{ - 1}{S_{[mt]}}f \stackrel {d}{\Rightarrow } Y(t),\] the convergence being understood in the sense of weak convergence of all finite dimensional distributions in $t$.