Generalizations of a theorem of Paul Erdős with application

Relying on Theorem 4, we prove the upper part of the law of iterated logarithm for uniformly bounded, equinormed, strongly multiplicative systems (ϕk(t):k=1,2,…;t∈[0,1]). We also state the central limit theorem for uniformly bounded multiplicative (ϕk(t)) systems.