Generalizations of a theorem of Paul Erdős with application

Let ( X k : k = 1 , 2 , … ) be a sequence of random variables. It is not assumed that the X k ʼs are mutually independent or that they are identically distributed. Set(1) S ( b , n ) : = ∑ k = b + 1 b + n X k and M ( b , n ) : = max 1 ⩽ k ⩽ n | S ( b , k ) | , where b ⩾ 0 and n ⩾ 1 are integers. We provide bounds on the expectation E M γ ( b , n ) in terms of given bounds of E | S ( b , n ) | γ , where γ > 1 is real. roblem 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 ( B 2 ) (see in Section 1). Our maximal Theorem 4 is a generalization of the Erdős-Stechkin maximal inequality (see both in Section 2). g on Theorem 4, we prove the upper part of the law of iterated logarithm for uniformly bounded, equinormed, strongly multiplicative systems ( ϕ k ( t ) : k = 1 , 2 , … ; t ∈ [ 0 , 1 ] ). We also state the central limit theorem for uniformly bounded multiplicative ( ϕ k ( t ) ) systems.