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Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

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Abstract

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ∪L p , based on the size of the norms of sums of sub-blocks of the firstn functions. The results are aplied to the study of a.e. convergence of series Σ n a n T n g/ n whenT is anL 2 -contraction,g∃L 2 , and {a n } is an appropriate sequence.

Given a sequence {f n }∪L p (Ω, μ), 1<p≤2, of independent centered random variables, we study conditions for the existence of a set ofx of μ-probability 1, such that for every contractionT on\(L_2 (\mathcal{Y},\pi )\) andg∈L 2 (π), the random power series Σ n f n (x)T n g converges π-a.e. The conditions are used to show that for {f n } centered i.i.d. withf 1∃L log+ L, there exists a set ofx of full measure such that for every contractionT on\(L_2 (\mathcal{Y},\pi )\) andg∃L 2 (π), the random series Σ n f n (x)T n g/n converges π-a.e.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, 84105, Beer Sheva, Israel

    Guy Cohen & Michael Lin

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  1. Guy Cohen
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  2. Michael Lin
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Correspondence to Guy Cohen.

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Dedicated to Hillel Furstenberg upon his retirement

We use Menchoff's own spelling of his name in the papers he wrote in French.

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Cohen, G., Lin, M. Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory. Isr. J. Math. 148, 41–86 (2005). https://doi.org/10.1007/BF02775432

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  • DOI: https://doi.org/10.1007/BF02775432

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