Complete stability of large order statistics
Let \(\{X_ n,n\geq 1\}\) be a sequence of i.i.d. random variables with common distribution function F(x) such that \(F(x)<1\) for all \(x\in R\) and let \(M_{nr}\) be the r th largest of \(X_ 1,...,X_ n\). The sequence \(\{M_{nr}\}_{n\geq r}\) is said to be \(\alpha\)-completely relatively stable for some \(\alpha\geq -1\) if \[ \sum^{\infty}_{n=r}n^{\alpha}P(| M_{nr}/a_ n-1| >\epsilon)<\infty \] for all \(\epsilon >0\) and some real sequence \(\{a_ n\}_{n\geq r}\). The authors show \(\alpha\)-complete relative stability of \(\{M_{nr}\}_{n\geq r}\) for some \(\alpha >-1\) and some \(r\geq 1\) iff \[ \int^{\infty}_{1}(1-F(\delta x))^{-t}dF(x)<\infty \text{ for all } 0<\delta <1 \] and some (equivalently, all) \(t>1\), and further, \(\alpha\)- complete relative stability for some \(\alpha >-1\) and some \(r\geq 1\) implies the same property for every \(\alpha >-1\) and every \(r\geq 1\).
scientific article; zbMATH DE number 4176246