The three-series theorem in additive arithmetical semigroups

Abstract. In this paper, we embed the additive arithmetical semigroup in a probability space \(\Omega := (\beta G,\sigma(\bar{\cal A}),\bar{\delta})\) where \(\beta G\) denote the Stone–Čech compactification of \(G\). We show that every additive function \(g\) on \(G\), \(g(a)=\sum\limits_{p^{k}\|a}g(p^{k}) \;\ (a\in G)\), can be identified with a sum \(\bar{g}=\sum\limits_{p}\bar{X_{p}}\) of independent random variables on \(\Omega\). The main result will be that the existence of the limit distribution of a real-valued additive function \(g\) is equivalent to the a.e. convergence of \(\bar{g}\).