Small Time Almost Sure Comparisons Between a Levy Process and its Maximal Jump Processes

"Small time" (as t down arrow 0) almost sure (a.s.) asymptotic comparisons between a Levy process X = (X-t)(t >= 0) and its maximal jump processes are studied, with the aim of delineating how much the process is influenced by its large jumps. The process may be comparable in size with its large jumps, or, alternatively, dominate them. X may even be replaced in an a.s. sense by its largest jump under certain conditions: it's possible to have X-t/ sup(0<s <= t) Delta X-s and/or X-t/ sup(0<s <= t) vertical bar Delta X-s vertical bar converging a.s. to 1 as t down arrow 0. Necessary and sufficient conditions in terms of the canonical measure of X are obtained for each of these kinds of behaviour; they correspond to a relatively mild singularity in the measure at 0. At the other extreme are cases when X dominates its maximal jump processes in the sense that, for example, vertical bar X-t vertical bar / sup(0<s <= t) vertical bar Delta X-s vertical bar diverges to infinity a.s. as t down arrow 0. The typical requirement here is that X be of bounded variation with non-zero drift