Weak Hölder convergence of processes with application to the perturbed empirical process

We consider stochastic processes as random elements in some spaces of Hölder functions vanishing at infinity. The corresponding scale of spaces ${\displaystyle C^{\alpha ,0}\_0}$ is shown to be isomorphic to some scale of Banach sequence spaces. This enables us to obtain some tightness criterion in these spaces. As an application, we prove the weak Hölder convergence of the convolution-smoothed empirical process of an i.i.d. sample ${\displaystyle (X_1,...,X_n)}$ under a natural assumption about the regularity of the marginal distribution function F of the sample. In particular, when F is Lipschitz, the best possible bound α<1/2 for the weak α-Hölder convergence of such processes is achieved.