A Central Limit Theorem under Contiguous Alternatives

In the statistical literature Le Cam's third lemma (cf. Hajek and Sidak (1967), page 208) is extensively used in order to get asymptotic normality of a statistic $S_n$ under contiguous alternatives from asymptotic normality of $S_n$ under the nullhypothesis. Since Le Cam's lemma utilizes the joint asymptotic normality of $S_n$ and $\log$-likelihood-ratio $\log L_n$, which is a sufficient but in general not a necessary condition for contiguity, it is not possible to get asymptotic normality of $S_n$ for all contiguous alternatives from this lemma. On the other hand one is interested in the limiting distribution of $S_n$ under all contiguous alternatives in order to get general power and efficiency results for the respective tests. In this paper we utilize a truncation method in order to prove asymptotic normality under all contiguous alternatives from asymptotic normality under the nullhypothesis for sums of independent random variables which are interesting in rank test theory, since they often are asymptotically equivalent to certain rank statistics under the nullhypothesis, and thus under contiguous alternatives, too.