A construction of a 𝜆-Poisson generic sequence

Abstract

Years ago Zeev Rudnick defined the λ \lambda -Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter λ \lambda . Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit λ \lambda -Poisson generic sequence over any alphabet and any positive λ \lambda , except for the case of the two-symbol alphabet, in which it is required that λ \lambda be less than or equal to the natural logarithm of 2 2 . Since λ \lambda -Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not λ \lambda -Poisson generic.