On a conjecture of Erdös about additive functions

For a real-valued additive function $f\colon \mathbb N \rightarrow \mathbb R$ and for each $n \in \mathbb N$ we define a distribution function \[ F_{n}(x):=\frac {1}{n}\#\{m\leq n\colon f(m)\leq x\}. \] In this paper we prove a conjecture of Erdös, which asserts that in order for the sequence $F_{n}$ to be (weakly) convergent, it is sufficient that there exist two numbers $a<b$ such that $\lim _{n\rightarrow \infty }(F_{n}(b)-F_{n}(a))$ exists and is positive.

The proof is based upon the use of the Stone–Čech compactification $\beta \mathbb N$ of $\mathbb N$ to mimic the behavior of an additive function as a sum of independent random variables.