  We analyse a trimmed stochastic process of the form ${}^{(r)}X_t= X_t -
\sum_{i=1}^r \Delta_t^{(i)}$, where $(X_t)_{t \geq 0}$ 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 $\bigl({}^{(r)}X_t, \Delta_t^{(r)}\bigr)$
in this case. We term this "large-trimming" behaviour. Concentrating on the
case $t=1$, we study joint convergence of $\bigl({}^{(r)}X_1,
\Delta_1^{(r)}\bigr)$ under linear normalization, assuming extreme
value-related conditions on the L\'evy 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 $\bigl({}^{(r)}X_1, \Delta_1^{(r)}\bigr)$ 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.
