Lipschitz Smoothness and Convergence with Applications to the Central Limit Theorem for Summation Processes

Abstract

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 $\lbrack 0, 1\rbrack^d$.