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

Hahn, Marjorie G.; Weiner, Daniel C.

J. Theor. Probab. 5, No. 1, 169-196 (1992).

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

Keywords:

self-normalization and studentization; weak convergence; nonnormal limits; asymptotic normality criterion; infinite random series

References:

[1]	Dudley, R. M. (1966). Convergence of Baire measures.Studia Math. XXVII, 251-268. · Zbl 0147.31301
[2]	Griffin, P. W., and Mason, D. M. (1991). On the asymptotic normality of self-normalized sums.Math. Proc. Camb. Phil. Soc. (to appear). · Zbl 0723.62008
[3]	Griffin, P. W., and Pruitt, W. E. (1987). The central limit problem for trimmed sums.Math. Proc. Camb. Phil. Soc. 102, 329-349. · Zbl 0631.60026
[4]	Hahn, M. G., Kuelbs, J., and Weiner, D. C. (1990). The asymptotic distribution of magnitude-winsorized sums via self-normalization.J. Theor. Prob. 3, 137-168. · Zbl 0696.60025
[5]	Hahn, M. G., Kuelbs, J., and Weiner, D. C. (1990). Self-normalization of censored sums and sums-of-squares in joint estimation of certain center and scale sequences.Ann. Prob. 18, 1284-1341. · Zbl 0725.62017
[6]	Hahn, M. G., and Weiner, D. C. (1992). Asymptotic behavior of self-normalized trimmed sums: nonnormal limits.Ann. Prob. (to appear). · Zbl 0749.60023
[7]	Mori, T. (1984). On the limit distributions of lightly trimmed sums.Math. Proc. Camb. Phil. Soc. 96, 507-516. · Zbl 0552.60018
[8]	Rudin, W. (1974).Real and Complex Analysis, 2nd ed., McGraw-Hill, New York. · Zbl 0278.26001
[9]	Taylor, A. (1985).General Theory of Functions and Integration. Dover, New York.