Low-Entropy Stochastic Processes for Generating k-Distributed and Normal Sequences, and the Relationship of These Processes with Random Number Generators †

Abstract

An infinite sequence x 1 x 2 . . . of letters from some alphabet { 0 , 1 , . . . , b - 1 } , b ≥ 2 , is called k-distributed ( k ≥ 1 ) if any k-letter block of successive digits appears with the frequency b - k in the long run. The sequence is called normal (or ∞-distributed) if it is k-distributed for any k ≥ 1 . We describe two classes of low-entropy processes that with probability 1 generate either k-distributed sequences or ∞-distributed sequences. Then, we show how those processes can be used for building random number generators whose outputs are either k-distributed or ∞-distributed. Thus, these generators have statistical properties that are mathematically proven.

A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

“…The representation of a real number by a continued fraction can be viewed as a source of information about the number. For this purpose, we need the notion of entropy, a rigorous tool at the crossroads of probability, information theory, and dynamical systems [2,3].…”