Almost sure weak convergence of the increments of Levy processes

Abstract

Let ( X t : t ≥ 0) be a stochastically continuous, real valued stochastic process with independent homogeneous increments, cadlag paths, X 0 = 0. We consider the behaviour, for fixed ω as h ↓ 0, of the increments (X t + h − X t) 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 L p ([0, 1], λ)(0 < p ≤ ∞) 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.