Limit theorems for trimmed sums
For \(n\geq 1\), let \(X^{(n)}_ 1\leq\cdots\leq X^{(n)}_ n\) be the order statistics from the first \(n\) observations of an i.i.d. sequence with distribution function \(F\), whose left-continuous inverse is \(Q\). Let \(h\) be a measurable function such that \(h\circ Q\) is square-integrable, and define, for \(n\geq 1\), the stochastic process \[ X_ n(t)=n^{- 1/2}\left\{\sum^{[nt]}_{j=[na]+1} h(X^{(n)}_ j)-n \int^ t_ a h\circ Q(x)dx\right\}. \] The limiting process of \(\{X_ n(t)\}\) as \(n\to\infty\) is determined explicitly. A similar study of the weighted trimmed sums \(\left\{\sum^{[nt]}_{j=1} J(j/n)X^{(n)}_ j, n\geq 1\right\}\) is also presented.
scientific article; zbMATH DE number 96934