A construction of a $λ$- Poisson generic sequence

Years ago Zeev Rudnick defined the $\lambda$-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 $\lambda$. Although it has long been known that almost all sequences, with respect to Lebesgue measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $\lambda$-Poisson generic sequence over an alphabet of at least three symbols, for any fixed positive real number $\lambda$. Since $\lambda$-Poisson genericity implies Borel normality, the constructed sequence is Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $\lambda$-Poisson generic.