  We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence
of integers {n_j}. Under a suitable integrability condition, we show that the
sequence converges to f a.e. Integral condition is only slightly larger than
what the sharp integrability condition would be, by a result of Konyagin. The
condition is: f is in L loglog L (logloglog L). The method of proof uses four
ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency
Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical
inequality of Zygmund, giving an improvement in the Hausdorff-Young inequality
for lacunary subsequences of integers, and (4) the extrapolation method of
Carro-Martin, which generalizes the work of Antonov and Arias-de-Reyna.
