Ratios of trimmed sums and order statistics
For i.i.d. \(X_ i\) let \(M_ n^{(n)}\leq\dots\leq M_ n^{(1)}\) be the ordered sample and \(X_ n^{(n)},\dots,X_ n^{(1)}\) be the sample arranged in the increasing order of modulus. Put \(S_ n=X_ 1+\dots+X_ n\), \(^{(r)}S_ n=S_ n-M_ n^{(1)}-\dots-M_ n^{(r)}\). \(^{(r)}\widetilde S_ n\) is defined by substituting \(M_ n^{(i)}\) by \(X_ n^{(i)}\) in \(^{(r)}S_ n\). \textit{R. A. Maller} and \textit{S. I. Resnick} [Proc. Lond. Math. Soc., III. Ser. 49, 385-422 (1984; Zbl 0525.60036)] and \textit{W. E. Pruitt} [Ann. Probab. 15, 885-896 (1987; Zbl 0625.60031)] studied i.i.d. \(X_ i\) for which \(^{(r)}S_ n\) and \(M_ n^{(r)}\) or \(^{(r)}\widetilde S_ n\) and \(X_ n^{(r)}\) are comparable in the sense that the corresponding ratios are bounded away from zero and infinity for large \(n\). In contrast to this the authors are interested in situations where large values of \(X\) are negligible compared with the sum. Roughly speaking this situation holds if \(X_ i\) has light tails whereas the situation studied by Maller and Resnick (loc. cit.) appears for heavy tails. Theorem 2.1 of the paper says that \(^{(r)}S_ n/M_ n^{(r)}\to\infty\) as \(n\to\infty\) holds iff the truncated mean \(\nu(x)=EXI(| X|<x)\) dominates \(xP(| X|>x)\), the truncated moment \(V(x)=EX^ 2I(| X|<x)\) and \((P(| X|\geq x)V(x))^{1/2}\) in the sense that the corresponding ratios tend to infinity. Surprisingly these conditions are independent of \(r\) so that the above convergence holds iff it is true for \(r=1\). The next two theorems concern the a.s. convergence of \(^{(r)}S_ n/X_ n^{(r)}\to\infty\) and \(^{(r)}S_ n/M_ n^{(r)}\to\infty\) for \(r=1,2,\dots\) and \(r=2,3,\dots\), respectively. Now necessary and sufficient conditions to the tails depend on \(r\) and it is shown by an example that the above mentioned equivalence is not true for a.s. convergence.
scientific article; zbMATH DE number 120264