The class of L-statistics is one of the most commonly used classes in statistical
inferences. We refer to monographs [5], [11], [12], [14] for the introduction to the theory of L-statistics. A survey on some modern applications of them in the economy and theory of actuarial risks can be found in [7]. There is an extensive literature on asymptotic properties of L-statistics, but its part concerning the large deviations is not so vast. We can mention a few of highly sharp results on this topic for L-statistics with smooth weight functions established in [13], [2], [1]. As to the trimmed L-statistics, the rst { and up to the recent time the single { result on probabilities of large deviations was obtained in [4], but under some strict and unnatural conditions.
Recently, the latter result was strengthened in [10], where a dierent approach than in [4] was proposed and implemented. In this note we present some of our recent results established in [8]-[9]. To conclude this short introduction, we want to mention a paper[3], and an interesting article [6], in which a general delta method in the theory of Cherno's type large deviations was proposed and illustrated by many examples including Mestimators and L-statistics.