Asymptotic expansions forL- andR-statistics under alternatives

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.