On the role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables

TL;DR

This paper investigates how ergodicity and mixing properties of the Casati-Prosen triangle map influence the distribution of sums of deterministic variables, revealing deviations from the classical central limit theorem and suggesting a Lorentzian-like limit distribution.

Contribution

It demonstrates that non-ergodic and non-mixing conditions lead to power-law distributions and that the sum distribution diverges from Gaussian behavior as the number of variables increases.

Findings

Ergodic and mixing maps produce Gaussian-like distributions.

Non-ergodic or non-mixing maps result in power-law distributions.

Sum distributions deviate from Gaussian as N increases, approaching a Lorentzian form.

Abstract

In this manuscript we analyse the behaviour of the probability density function of the sum of $N$ deterministic variables generated from the triangle map of Casati-Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. When these properties are modified the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables $N$ increases the distance to Gaussian distribution increases. This behaviour goes against standard central limit theorem. By extrapolation of our finite size results we preview that in the limit of $N$ going to infinity the distribution has the same asymptotic decay as a Lorenztian (or a $q=2$-Gaussian).