Strong laws at zero for trimmed Lévy processes

Abstract

We study the almost sure (a.s.) behaviour of a Lévy process $(X_t)_{t\ge 0}$ on $\mathbb{R}$ with extreme values removed, giving necessary and sufficient conditions for the a.s. convergence as $t\to0$ of normed and centered versions of "trimmed" processes, in which the $r$ largest positive jumps or the $r$ largest jumps in modulus of $X$ up to time $t$ are subtracted from it. Integral criteria in terms of the canonical measure of $X$ are given for the required convergences, under natural conditions on the norming functions. Random walk results of Mori (1976, 1977) and Lévy process results of Shtatland (1965) and Rogozin (1968) are thereby generalised. Another application is to characterise the relative stability at 0 of the trimmed processes, in probability and almost surely.<br />