We prove that certain jump summation processes converge in distribution for the uniform topology to the Brownian sheet, while smoothed summation processes converge for various Lipschitz topologies. These results follow after a careful study of abstract, generalized Lipschitz spaces. Along the way we affirm a conjecture about smoothness and continuity of processes defined on $\lbrack 0, 1\rbrack^d$.
"Lipschitz Smoothness and Convergence with Applications to the Central Limit Theorem for Summation Processes." Ann. Probab. 9 (5) 831 - 851, October, 1981. https://doi.org/10.1214/aop/1176994311