  We adapt the techniques in Stigler [Ann. Statist. 1 (1973) 472--477] to
obtain a new, general asymptotic result for trimmed $U$-statistics via the
generalized $L$-statistic representation introduced by Serfling [Ann. Statist.
12 (1984) 76--86]. Unlike existing results, we do not require continuity of an
associated distribution at the truncation points. Our results are quite general
and are expressed in terms of the quantile function associated with the
distribution of the $U$-statistic summands. This approach leads to improved
conditions for the asymptotic normality of these trimmed $U$-statistics.
