We present a solution to the goodness-of-fit problem for multivariate observations, using the innovation process for the (sequential) empirical distribution function with respect to a conveniently chosen linear ordering or scanning system in $R^m$.
"An Innovation Approach to Goodness-of-Fit Tests in $R^m$." Ann. Statist. 16 (4) 1503 - 1516, December, 1988. https://doi.org/10.1214/aos/1176351051