Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits. II. (English) Zbl 0743.60024

Summary: Let \(\{X_ j\}\) be independent, identically distributed random variables which are symmetric about the origin and have a continuous nondegenerate distribution \(F\). Let \(\{X_ n(1),\dots,X_ n(n)\}\) denote the arrangement of \(\{X_ 1,\dots,X_ n\}\) in decreasing order of magnitude, so that with probability one, \(| X_ n(1)| > | X_ n(2)| >\dots>| X_ n(n)|\). For integers \(r_ n\to\infty\) such that \(r_ n/n\to 0\), define the self-normalized trimmed sum \(T_ n=\sum^ n_{i=r_ n}X_ n(i)/\{\sum^ n_{i=r_ n}X^ 2_ n(i)\}^{1/2}\). In part I [Ann. Probab. 20, No. 1, 455-482 (1992)] it is shown that under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion for \(T_ n\), various nonnormal limit laws for \(T_ n\) arise which are represented by means of infinite random series. The analytic condition is now extended and the previous approach is refined to obtain limits which are mixtures of a normal, a Rademacher, and a law represented by a more general random series. Each such limit law actually arises as can be seen from the construction of a single distribution whose corresponding \(\{{\mathcal L}(T_ n)\}\) generates all of the laws along different subsequences, at least if \(\{r_ n\}\) grows sufficiently fast. Another example clarifies the limitations of the basic approach.


60F05 Central limit and other weak theorems
60B10 Convergence of probability measures
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