Trimmed L\'evy Processes and their Extremal Components

We analyse a trimmed stochastic process of the form ${}^{®}X_t= X_t - \sum_{i=1}^r \Delta_t^{(i)}$,where$(X_t)_{t \geq 0}$ is a driftless subordinator on $\mathbb{R}$ with its jumps on $[0,t]$ ordered as ${\mathrm{\Delta }}_t^{(1)}\ge {\mathrm{\Delta }}_t^{(2)}\cdots$. When $r\to \mathrm{\infty }$, both ${}^{®}X_t \to 0$and$\Delta_t^{®} \to 0$a\mathrm{.}s\mathrm{.}foreach$t>0$, and it is interesting to study the weak limiting behaviour of $({}^{(r)}{X}_t,{\mathrm{\Delta }}_t^{(r)})$ in this case. We term this “large-trimming” behaviour. Concentrating on the case $t=1$, we study joint convergence of $\bigl({}^{®}X_1, \Delta_1^{®}\bigr)$ under linear normalization, assuming extreme value-related conditions on the L'evy measure of $X$ which guarantee that ${\mathrm{\Delta }}_1^{(r)}$ has a limit distribution with linear normalization. Allowing ${}^{(r)}{X}_1$ to have random centering and scaling in a natural way, we show that $({}^{(r)}{X}_1,{\mathrm{\Delta }}_1^{(r)})$ has a bivariate normal limiting distribution, as $r\to \mathrm{\infty }$; but replacing the random normalizations with natural deterministic ones produces non-normal limits which we can specify.