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

We prove extensions of Menchoff's inequality and the MenchoffRademacher theorem for sequences {f_n} ⊂ L_p, 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/n^α when T is an L ₂-contraction, g ε L₂, and {a_n} is an appropriate sequence. Given a sequence {f_n} 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 f_n(x)Tⁿg converges π-a.e. The conditions are used to show that for {f_n} 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.