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.