Small-time versions of Strassen's law for Lévy processes

We study aspects of the 'small-time' behaviour (as t ↓ 0) of a Lévy process X(t), obtaining a very general small-time version of Strassen's almost sure (a.s.) functional law of the iterated logarithm (LIL) for random walks. The class of Lévy processes for which this holds is characterised by an explicit analytic condition on the Lévy measure of X, related to an analogous condition of Kesten for a generalised (large-time) random walk LIL. Both centred and uncentred versions of the small-time result are proved. Subsidiary results concerning functional weak convergence of X(t) to Brownian motion as t ↓ 0 are shown to be equivalent to the main a.s. results. The quadratic variation process of X is considered, and applications via continuous functionals are suggested.