Some limit theorems of almost periodic function systems under the relative measure (Q1118248)

\textit{M. Kac} and \textit{H. Steinhaus} [Stud. Math. 7, 1-15 (1938; Zbl 0018.07601)] proved that if a real sequence \(\{\lambda_ j\}\) is algebraically independent, then the sequence \(\{\) \(\sqrt{2} \cos \lambda_ jx\}\) obeys the CLT. The first aim of the present author is to weaken the condition on \(\{\lambda_ j\}\) by introducing the notion of signed sum condition. Second, he proves a SLLN. Third, he proves a functional CLT of Prohorov type and derives other limit theorems. Finally, he proves a functional (i.e. Strassen type) LIL. These results are treated in the framework of uniformly bounded (strongly or equinormed) multiplicative systems defined by \textit{G. Alexits} [Convergence problems of orthogonal series (1961; Zbl 0098.274)].