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)} subset of 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 Sigma(n) a(n)T(n)g/n(alpha) when T is an L-2-contraction, g is an element of L-2, and {a(n)} is an appropriate sequence. Given a sequence {f(n)} subset of L-p(Omega, mu), 1 < p <= 2, of independent centered random variables, we study conditions for the existence of a set of x of mu-probability 1, such that for every contraction T on L-2 (Y, pi) and g is an element of L-2(pi), the random power series Sigma(n) f(n)(x)T(n)g converges pi-a.e. The conditions are used to show that for {f(n)} centered i.i.d. with f(1) is an element of L log(+) L, there exists a set of x of full measure such that for every contraction T on L-2 (Y, pi) and g is an element of L-2 (pi), the random series Sigma(n)f(n)(x)T-n g/n converges pi-a.e.