For $U$-Statistics and von Mises' differentiable statistical functions, when the regular functional is stationary of order zero, almost sure convergence to appropriate Wiener processes is studied. A second almost sure invariance principle, particularly useful in the context of the law of iterated logarithm and the probability of moderate deviations, is also established.
"Almost Sure Behaviour of $U$-Statistics and Von Mises' Differentiable Statistical Functions." Ann. Statist. 2 (2) 387 - 395, March, 1974. https://doi.org/10.1214/aos/1176342675