The compactness of distributions of a sequence of additive functions (Q1098880)

Let \(f_ n(m)\), \(n=1,2,..\). be a sequence of real-valued additive functions and \[ F_ n(z)=(1/n) \#\{m\leq n: f(n)-\alpha (n)<z\} \] their distributions with the center \(\alpha(n)\). The principal problem of probabilistic number theory is to find conditions for the weak convergence of \(F_ n\) to some nondegenerate limit law. No satisfactory necessary and sufficient condition is known even in the most important case when \(f_ n(m)=f(m)/B(n)\), a single function with a normalization. The related problem of convergence to a degenerate law (law of large numbers) was completely solved by the reviewer [Stud. Sci. Math. Hung. 14, 247--253 (1979; Zbl 0486.10044)]. The author presents a complete solution to the problem when this sequence \((F_ n)\) of distributions is conditionally compact (CC). Take a sequence \(\mu(n)\) of real numbers and let \(\xi_{np}\) (\(p\) prime) be independent random variables assuming the values \(f_ n(p^ r)-\mu (n)\log p^ r\) with probability \(p^{-r}(1-1/p),\) \(r=0,1,..\). The sequence \((F_ n)\) is CC for a suitable choice of \(\alpha(n)\) if and only if the distribution of the variables \(\sum_{p\leq n}\xi_{np}-\beta(n)\) is CC for a suitable choice of \(\beta(n)\) and a bounded \(\mu(n)\). Another equivalent condition is given directly in terms of the \(f_ n(p^ r).\) Since a sequence is convergent if and only if it is CC and has at most one limit point, this result can be applied to the problem of limiting laws, which is done in Theorem 3 of the paper, while Theorem 4 applies it to the study of possible norming functions that can appear in the limiting law of a single additive function.