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

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 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 andg∃L ₂ (π), the random series Σ _n f _n (x)T ⁿ g/n converges π-a.e. We use Menchoff's own spelling of his name in the papers he wrote in French. Dedicated to Hillel Furstenberg upon his retirement