The almost sure behavior of the oscillation modulus of the multivariate empirical process

Let [omega]n denote the oscillation modulus of the uniform multivariate empirical process, defined as the variation of the process over multi-dimensional intervals with Lebesgue measure not exceeding an [epsilon] (0,1). The a.s. limiting behavior of [omega]n is established for sequences {an} of five different orders of magnitude that constitute an essentially complete spectrum of possibilities. Extensions to processes with underlying d.f. more general than the uniform are indicated. The results may have applications in density estimation and in the theory of multivariate spacings. For related results in the univariate case we refer in particular to Mason, Shorack and Wellner (1983) and Mason (1984), and for a general setting to Alexander (1984).