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The asymptotic distribution of trimmed sums. (English) Zbl 0647.62030
Let \(X_{1,n}\leq \cdot \cdot \cdot X_{n,n}\) be the order statistics of n independent random variables with a common distribution function F, and let m(n) and k(n) be positive integers such that m(n)\(\to \infty\), k(n)\(\to \infty\), m(n)/n\(\to 0\) and k(n)/n\(\to 0\) as \(n\to \infty\). The authors give a necessary and sufficient condition for the existence of normalizing constants \(A_ n>0\) and \(B_ n\) such that the sequence \[ T_ n=A_ n^{-1}\{\sum^{n-k(n)}_{i=m(n)+1}X_{i,n}-B_ n\} \] is stochastically compact. They show that if a subsequence \(\{T_{n'}\}\) converges to V in distribution, then V is represented as follows: \[ V=\int_{0}^{-Z_ 1}(Z_ 1+x)d\psi_ 1(x)+Z+\int^{0}_{-Z_ 2}(Z_ 2+x)d\psi_ 2(x), \] where \(\psi_ 1\) and \(\psi_ 2\) are nondecreasing functions determined by m(n), k(n) and the tail of F, and \((Z_ 1,Z,Z_ 2)\) is a trivariate normal random vector.
They also give a necessary and sufficient condition for the existence of \(A_ n\) and \(B_ n\) such that \(T_ n\) is asymptotically normal. A variant of Stigler’s theorem when m(n)/n\(\to \alpha\), and k(n)/n\(\to 1- \beta\), where \(0<\alpha <\beta <1\), is also obtained.
Reviewer: T.Mori

62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
62G30 Order statistics; empirical distribution functions
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