## 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.

### MSC:

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

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