Hahn, M. G.; Kuelbs, J.; Samur, J. D. Asymptotic normality of trimmed sums of \(\Phi\)-mixing random variables. (English) Zbl 0639.60028 Ann. Probab. 15, 1395-1418 (1987). Let \(\{X_ n\}\) be a strictly stationary sequence of Hilbert-space- valued random variables which is \(\Phi\)-mixing. Set \(S_ n=X_ 1+...+X_ n\) and for constants \(j_ n\), \(k_ n\) let \(S_ n(j_ n,k_ n)\) denote the partial sum \(S_ n\) reduced by those terms corresponding to the \(j_ n\) largest values of \((\| X_ 1\|,...,\| X_ n\|)\) provided that they exceed the threshold \(k_ n\) in norm. This paper investigates conditions which are sufficient to ensure that the sequence \(S_ n(j_ n,k_ n)\) suitably scaled and normed is tight and all weak limits of subsequences are centred Gaussian. The results are related to a conjecture of Ibragimov [I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables (1971; Zbl 0154.422)] that if the random variables are real-valued with E \(X_ j=0\), \(E(X^ 2_ j)=1\) and \(E(S^ 2_ n)\to \infty\) then \(S_ n\) can be normalized to a non-degenerate Gaussian limit. The authors give conditions that guarantee that once the maximal terms of \(| X_ 1|,...,| X_ n|\) are deleted from \(S_ n\) in an appropriate way the only possible non-degenerate limits are Gaussian. Reviewer: D.P.Kennedy Cited in 6 Documents MSC: 60F05 Central limit and other weak theorems 60F15 Strong limit theorems 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) Keywords:asymptotic normality; central limit theorem; strictly stationary sequence of Hilbert-space-valued random variables; \(\Phi \)-mixing Citations:Zbl 0154.422 PDFBibTeX XMLCite \textit{M. G. Hahn} et al., Ann. Probab. 15, 1395--1418 (1987; Zbl 0639.60028) Full Text: DOI