Stable Functional CLT for deterministic systems

TL;DR

This paper demonstrates that alpha stable Lévy motions can be simulated by ergodic, aperiodic measure-preserving transformations, establishing a functional CLT for deterministic systems across different alpha regimes.

Contribution

It provides a novel method to simulate alpha stable Lévy motions using deterministic ergodic transformations, extending the classical CLT to a broader class of systems.

Findings

Simulation of alpha stable Lévy motions via ergodic transformations

Convergence of partial sums to stable Lévy motions for various alpha

Existence of functions with time series in the domain of attraction of stable laws

Abstract

We show that alpha stable L\'evy motions can be simulated by any ergodic and aperiodic probability preserving transformation. Namely we show: - for $0<\alpha<1$ and every $\alpha$ stable L\'evy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$. - for $1\leq \alpha <2$ and every symmetric alpha stable L\'evy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$, - for $1< \alpha <2$ and every $-1\leq\beta \leq 1$ 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.