A construction of a λ- Poisson generic sequence
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Years ago Zeev Rudnick defined the λ-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. 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 λ-Poisson generic sequence over an alphabet of at least three symbols, for any fixed positive real number λ. Since λ-Poisson genericity implies Borel normality, the constructed sequence is Borel normal. The same construction provides explicit instances of Borel normal sequences that are not λ-Poisson generic.
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