Kubilius-type sequences of additive functions
A classical result of Kubilius asserts that an additive function belonging to his class \(H\) has a limiting distribution if and only if the associated sum formed from related independent random variables does. Here a partial converse is proved. Let \(f_x\) be a family of integer-valued strongly additive functions, \(\nu _x\) the distribution of \(f_x\) on \(\{1, \dots , x\}\), and let \(P_x\) be the distribution of the naturally associated sum of independent variables. Under the (probably not very essential) additional assumption that \(f_x(p)\) is uniformly bounded, the authors give a necessary and sufficient condition for the validity of a local limit law in the form \[ \sup _k \nu _x(f_x(n)=k)-P_x(k) \to 0 . \] Write \(E(x) = \sum _{p\leq x, f_x(p)\neq 0} 1/p \). The condition is essentially that we can split the \(x\)'s into two subsequences, \(E(x)\to \infty \) on one and the functions are in the Kubilius class on the other. If \(E(x)\to \infty \), then the concentration of both \(f_x\) and \(P_x\) tends to 0, so the limiting law is a trivial one. A stronger measure could be obtained by replacing the supremum by a sum; it will be probably difficult to clarify this kind of convergence.
scientific article; zbMATH DE number 5076499
scientific article