Generalized one-sided laws for the larger observations of a triangular array (Q1866073)

Consider a triangular array of rowwise i.i.d. random variables with common density \(f(x)=px^{p-1}I\) \((x\geq 1)\), where \(p>1\). The author derives some generalized (weak and strong) laws of the iterated logarithm for the weighted sums of order statistics \(\sum^N_{n=k} n^\alpha X_{nk}\), as \(N\to \infty\), where \(X_{nk}\) denotes the \(k\)th largest order statistic from the \(n\)th row of the array, and \(pk=1\), \(\alpha+ k+1>0\). It is a borderline situation in which the random variables under consideration are neither i.i.d. nor possess a first moment, and thus lead to rather unusual limit theorems.