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Proceedings of the London Mathematical Society Advance Access originally published online on October 7, 2008
Proceedings of the London Mathematical Society 2009 98(2):531-558; doi:10.1112/plms/pdn041
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© 2008 London Mathematical Society

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

R. A. Maller

Centre for Mathematics and its Applications
Australian National University
Canberra, ACT 0200
Australia
and
School of Finance and Applied Statistics
Australian National University
Canberra, ACT 0200
Australia

Received 27 March 2007. Revision received 30 January 2008.

We study aspects of the ‘small-time’ behaviour (as t {downarrow} 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 {downarrow} 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.

2000 Mathematics Subject Classification 60G51, 60F15, 60F17, 60F05 (primary), s60J65, 60J75 (secondary).

This research was partially supported by ARC grant DP0664603.


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