The convergence to the Poisson law. I: Integer-valued additive functions
Let \(f_x\) be a sequence of integer-valued nonnegative additive function, satisfying, as \(x\to\infty\), \(f_x(p^k) \to 0\) for each prime-power, \(\max_{p\leq x}|f_x(p) |= O(1)\) and \(\sum_{\sqrt p\leq x,f_x(p) \neq 0} 1/p \to 0\). Under these assumptions a necessary and sufficient condition is found for the frequencies \(\nu_x(f_x(n)<u)\) to converge to a Poisson law. For a standard Poisson law with parameter \(\lambda\) this is \[ \sum_{p^r\leq x, f_x (p^r)=1} p^{-r} \to\lambda, \quad \sum_{p^r\leq x,f_x(p^r) >1} p^{-r}\to 0, \quad \sum_{p^r\leq x, f_x (p^r) \geq 1} p^{-r} \log p^r= o(\log x). \] The reviewer feels that the second preliminary assumption is probably unnecessary. The third is essential; if it is dropped, a wide class of new examples arises which seems to be difficult to describe. The first is connected with the unsolved problem of whether a fixed additive function can have a nondegenerate Poisson distribution [see \textit{P. D. T. A. Elliott}, Probabilistic number theory, Vol. II: Central limit theorems, Springer (1980; Zbl 0431.10030), pp. 172, 337].
scientific article; zbMATH DE number 939535