Strong moderate deviation theorems. (English) Zbl 0757.60015

Inglot, Tadeusz; Kallenberg, Wilbert C. M.; Ledwina, Teresa

Ann. Probab. 20, No. 2, 987-1003 (1992).

Let \(T_ n\) be a sequence of random variables such that there are r.v. \(W_ n\) satisfying the following conditions \[ P(\sqrt n| T_ n-W_ n|>c_ 1\lg n+x)\leq c_ 2 e^{-c_ 3x} \] for all large enough \(n\) and all \(0<x<c_ 4 n^{1/3}\) and assume that \[ P(W_ n>x)=\exp(-2^{- 1} ax^ 2+g(x)) \] with \(g(x)\) satisfying certain regularity assumptions. The authors show that then
\(\lim_{n\to\infty}(P(T_ n>x_ n)/P(W_ n>x_ n))=1\), if \(x_ n\to\infty\) and \(x_ n=o(n^{-1/6})\). The result is applied to weighted Kolmogorov-Smirnov statistics, generalized Cramér-von Mises statistics, chi-square statistics, Watson statistics, quadratic statistics and \(L\)- statistics.

Reviewer: W.Bryc (Cincinnati)

MSC:

60F10	Large deviations
60G30	Continuity and singularity of induced measures

Keywords:

moderate deviations; Kolmogorov-Smirnov statistics; Cramér-von Mises statistics; chi-square statistics; Watson statistics; quadratic statistics; \(L\)-statistics