Trimmed Lévy Processes and their Extremal Components

Abstract

We analyse a trimmed stochastic process of the form ${}^{(r)}{X}_t=X_t-{\sum }_{i=1}^r{\Delta }_t^{(i)}$, where $(X_t{)}_{t\ge 0}$ is a driftless subordinator on $\mathbb{R}$ with its jumps on $[0,t]$ ordered as ${\Delta }_t^{(1)}\ge {\Delta }_t^{(2)}\cdots$. When $r\to \infty$, both ${}^{(r)}{X}_t\to 0$ and ${\Delta }_t^{(r)}\to 0$ a.s. for each $t>0$, and it is interesting to study the weak limiting behaviour of $({}^{(r)}{X}_t,{\Delta }_t^{(r)})$ in this case. We term this "large-trimming" behaviour. Concentrating on the case $t=1$, we study joint convergence of $({}^{(r)}{X}_1,{\Delta }_1^{(r)})$ under linear normalization, assuming extreme value-related conditions on the Lévy measure of $X$ which guarantee that ${\Delta }_1^{(r)}$ has a limit distribution with linear normalization. Allowing ${}^{(r)}{X}_1$ to have random centering and scaling in a natural way, we show that $({}^{(r)}{X}_1,{\Delta }_1^{(r)})$ has a bivariate normal limiting distribution, as $r\to \infty$; but replacing the random normalizations with natural deterministic ones produces non-normal limits which we can specify.

19 pages