Generalizations of a theorem of Paul Erdős with application

Abstract

Let (Xk:k=1,2,…) be a sequence of random variables. It is not assumed that the Xkʼs are mutually independent or that they are identically distributed. Set(1)S(b,n):=∑k=b+1b+nXkandM(b,n):=max1⩽k⩽n|S(b,k)|, where b⩾0 and n⩾1 are integers. We provide bounds on the expectation EMγ(b,n) in terms of given bounds of E|S(b,n)|γ, where γ>1 is real.This problem goes back to a theorem of Erdős [P. Erdős, On the convergence of trigonometric series, J. Math. Phys. (Massachusetts Institute of Technology), 22 (1943), 37–39] on the almost everywhere convergence of such trigonometric series that the indices of the nonzero coefficients satisfy condition (B2) (see in Section 1). Our maximal Theorem 4 is a generalization of the Erdős-Stechkin maximal inequality (see both in Section 2).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.