  We prove that, for functions in the Orlicz class LloglogLloglogloglogL,
lacunary subsequences of the Fourier and the Walsh-Fourier series converge
almost everywhere. Our integrability condition is less stringent than the
homologous assumption in the almost everywhere convergence theorems of Lie
(Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple
logarithmic term is replaced by a triple logarithm. Our proof of the
Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's
argument, avoids the use of Antonov's lemma, arguing directly via novel
weak-L^p bounds for the Walsh-Carleson operator.
