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

Asymptotic distributions for a class of generalized $L$-statistics