Griffin, Philip S.; Mason, David M. On the asymptotic normality of self-normalized sums. (English) Zbl 0723.62008 Math. Proc. Camb. Philos. Soc. 109, No. 3, 597-610 (1991). Summary: Let \(X_ 1,...,X_ n\) be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let \(S_ n(r_ n)\) denote their sum when the \(r_ n\) largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the Studentized version of \(S_ n(r_ n)\), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when \(r_ n=r\) these conditions are the same, but when \(r_ n\to \infty\), the former holds more generally. Cited in 21 Documents MSC: 62E20 Asymptotic distribution theory in statistics Keywords:modulus trimmed sum; necessary and sufficient conditions; asymptotic normality; Studentized version; scalar normalized version PDFBibTeX XMLCite \textit{P. S. Griffin} and \textit{D. M. Mason}, Math. Proc. Camb. Philos. Soc. 109, No. 3, 597--610 (1991; Zbl 0723.62008) Full Text: DOI References: [1] DOI: 10.2307/2286068 · Zbl 0188.50304 [2] DOI: 10.2307/1990824 · Zbl 0047.37502 [3] Cs?rg?, Sums, Trimmed Sums and Extremes (1990) [4] Bingham, Regular Variation (1987) [5] Gnedenko, Limit Distributions for Sums of Independent Random Variables (1954) · Zbl 0056.36001 [6] DOI: 10.1214/aop/1176996846 · Zbl 0272.60016 [7] Mori, Math. Proc. Cambridge Philos. Soc 96 pp 507– (1984) [8] DOI: 10.1007/BF01063331 · Zbl 0696.60025 [9] Griffin, Math. Proc. Cambridge Philos. Soc 102 pp 329– (1987) [10] DOI: 10.2307/1426784 · Zbl 0469.60025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.