Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System

TL;DR

This paper investigates the limit distributions of time averages in infinite measure dynamical systems, revealing convergence to generalized arcsine and stable distributions, which implies intrinsic randomness and non-decaying correlations.

Contribution

It establishes new limit theorems showing convergence to generalized arcsine and stable laws for non-$L^1$ observation functions in infinite measure systems.

Findings

Time averages converge to the generalized arcsine distribution.

When the observation function has infinite mean, convergence to a stable distribution is observed.

Correlation functions are intrinsically random and do not decay.

Abstract

Limit theorems for the time average of some observation functions in an infinite measure dynamical system are studied. It is known that intermittent phenomena, such as the Rayleigh-Benard convection and Belousov-Zhabotinsky reaction, are described by infinite measure dynamical systems.We show that the time average of the observation function which is not the $L^1(m)$ function, whose average with respect to the invariant measure $m$ is finite, converges to the generalized arcsine distribution. This result leads to the novel view that the correlation function is intrinsically random and does not decay. Moreover, it is also numerically shown that the time average of the observation function converges to the stable distribution when the observation function has the infinite mean.