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 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 ${\lambda}$-Poisson generic sequence over any alphabet and any positive ${\lambda}$, except for the case of the two-symbol alphabet, in which it is required that ${\lambda}$ be less than or equal to the natural logarithm of $2$. Since ${\lambda}$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not ${\lambda}$-Poisson generic.

翻译：多年前，Zeev Rudnick 将${\lambda}$-泊松生成序列定义为有限字母表上的无限符号序列，其初始段中长词的出现次数服从参数为${\lambda}$的泊松分布。尽管几乎所有序列（在均匀测度下）都是泊松生成的，但至今尚未给出显式实例。本文针对任意字母表及任意正数${\lambda}$，给出了一个显式${\lambda}$-泊松生成序列的构造方法，但在双符号字母表的情形下，要求${\lambda}$不大于$2$的自然对数。由于${\lambda}$-泊松生成性蕴含Borel正规性，所构造的序列均为Borel正规序列。该构造还提供了非${\lambda}$-泊松生成的Borel正规序列的显式实例。