Deviations from Gaussianity in deterministic discrete time dynamical systems

TL;DR

This paper investigates how deterministic discrete-time systems deviate from Gaussian behavior, using Edgeworth expansions to describe these deviations and providing explicit formulas and numerical validation.

Contribution

It introduces explicit asymptotic expansions for deviations from Gaussianity in deterministic systems, validated by numerical evidence.

Findings

Edgeworth expansions accurately describe deviations from normality

Explicit formulas for asymptotic deviations are derived

Numerical results confirm theoretical predictions

Abstract

In this paper we examine the deviations from Gaussianity for two types of random variable converging to a normal distribution, namely sums of random variables generated by a deterministic discrete time map and a linearly damped variable driven by a deterministic map. We demonstrate how Edgeworth expansions provide a universal description of the deviations from the limiting normal distribution. We derive explicit expressions for these asymptotic expansions and provide numerical evidence of their accuracy.