Classical limit theorems and high entropy MIXMAX random number generator

TL;DR

This paper explores how deterministic Anosov C-systems, exemplified by the MIXMAX pseudorandom number generator, satisfy classical probability limit theorems due to their ergodic and mixing properties.

Contribution

It establishes a theoretical link between classical probability limit theorems and the behavior of deterministic chaotic systems like MIXMAX.

Findings

MIXMAX satisfies laws of large numbers

MIXMAX adheres to the central limit theorem

MIXMAX exhibits properties consistent with the law of the iterated logarithm

Abstract

We investigate the interrelation between the distribution of stochastic fluctuations of independent random variables in probability theory and the distribution of time averages in deterministic Anosov C-systems. On the one hand, in probability theory, our interest dwells on three basic topics: the laws of large numbers, the central limit theorem and the law of the iterated logarithm for sequences of real-valued random variables. On the other hand we have chaotic, uniformly hyperbolic Anosov C-systems defined on tori which have mixing of all orders and nonzero Kolmogorov entropy. These extraordinary ergodic properties of deterministic Anosov C-systems ensure that the above classical limit theorems for sums of independent random variables in probability theory are fulfilled by the time averages for the sequences generated by the C-systems. The MIXMAX generator of pseudorandom numbers represents the C-system for which the classical limit theorems are fulfilled.