Stable Functional CLT for deterministic systems

Abstract

We show that alpha stable Lévy motions can be simulated by any ergodic and aperiodic probability preserving transformation. Namely we show: - for $0<\alpha <1$ and every $\alpha$ stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$. - for $1\le \alpha <2$ and every symmetric alpha stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$, - for $1<\alpha <2$ and every $-1\le \beta \le 1$ there exists a function f whose associated time series is in the classical domain of attraction of an $S_{\alpha }(\ln (2),\beta ,0)$ random variable.

36 pages