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

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).

arXiv comment14 pages. To appear in Physics Letters A

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Cited by in corpus (5)

• Central Limit Behavior in the Kuramoto model at the 'Edge of Chaos'
• Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasi-periodic edge of chaos
• Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem
• Generalized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions
• On ergodic and mixing properties of the triangle map