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- W2000915092 abstract "Abstract 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. 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 ( B 2 ) (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 ( ϕ k ( t ) : k = 1 , 2 , … ; t ∈ [ 0 , 1 ] ). We also state the central limit theorem for uniformly bounded multiplicative ( ϕ k ( t ) ) systems." @default.
- W2000915092 created "2016-06-24" @default.
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- W2000915092 date "2013-09-01" @default.
- W2000915092 modified "2023-09-27" @default.
- W2000915092 title "Generalizations of a theorem of Paul Erdős with application" @default.
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- W2000915092 doi "https://doi.org/10.1016/j.endm.2013.07.064" @default.
- W2000915092 hasPublicationYear "2013" @default.
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