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 MenchoffRademacher theorem for sequences {fn} ⊂ Lp, based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series ∑n a nTng/nα when T is an L 2-contraction, g ε L2, and {an} is an appropriate sequence. Given a sequence {fn} sub; L P(Ω,μ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of μ-probability 1, such that for every contraction T on L 2(γ,π) and g ε L2(π), the random power series ∑n fn(x)Tng converges π-a.e. The conditions are used to show that for {fn} centered i.i.d. with f 1 ε L log+ L, there exists a set of x of full measure such that for every contraction T on L2(γ, π) and g ε L2(π), the random series ∑n fn(x)T np/n converges π-a.e.

Original languageEnglish
Pages (from-to)41-86
Number of pages46
JournalIsrael Journal of Mathematics
Volume148
DOIs
StatePublished - 1 Dec 2005

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