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Asymptotic normality of trimmed sums of \(\Phi\)-mixing random variables. (English) Zbl 0639.60028

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

MSC:

60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)

Citations:

Zbl 0154.422
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