Chentsov-Billingsley type fluctuation inequalities for stochastic processes whose time parameter ranges over the $q$-dimensional unit cube are derived and used to establish weak convergence results for such processes.
P. J. Bickel. M. J. Wichura. "Convergence Criteria for Multiparameter Stochastic Processes and Some Applications." Ann. Math. Statist. 42 (5) 1656 - 1670, October, 1971. https://doi.org/10.1214/aoms/1177693164