Almost sure weak convergence of the increments of Lévy processes

Let (Xt : t >= 0) be a stochastically continuous, real valued stochastic process with independent homogeneous increments, cadlag paths, X0 = 0. We consider the behaviour, for fixed [omega] as h [downwards arrow] 0, of the increments (Xt + h - Xt)/a(h) as a function of t in [0, 1] with Lebesgue measure, a(·) belonging to some natural class of functions. Generally speaking, it is not possible to find a(·) so that almost surely the normalized increments have a non-trivial limit in Lp([0, 1], [lambda])(0 < p <= [infinity]) or pointwise. However it is possible to give necessary and sufficient conditions on the process so that for almost every path the normalized increments have a non-trivial limit in the sense of weak convergence of distributions, for an appropriate choice of a(·). This extends a previous result for the Wiener process. The result holds if one replaces Lebesgue measure on [0, 1] by an absolutely continuous random measure.