The range of almost sure limits of $F$-variation for a class of Gaussian random fields is considered by adopting a class of sequences of partitions in the parameter space of the random field. The application to Levy's Brownian motion explains, in the case of two-dimensional parameters, that the almost sure limit given by Berman is the maximum in a range.
"Maximum in the Levy-Baxter Theorem for Gaussian Random Fields." Ann. Probab. 7 (1) 173 - 178, February, 1979. https://doi.org/10.1214/aop/1176995161