Complete convergence for arrays of minimal order statistics (Q1777695)

Let \(\{X_{nk}:\,1\leq k\leq m_n,\,n\geq1\}\) be an array of row-wise independent Pareto distributed random variables with density \(p_nx^{-p_n-1}\), \(x\geq1\), and let \(X_{n(k)}\) denote its \(k\)\,th smallest order statistic from the \(n\)\,th row. If \(p_n(m_n-k+1)>1\) then the expectation \(E(X_{n(k)})\) exists and hence classical strong laws of large numbers can be obtained. In the case \(p_n(m_n-k+1)<1\) it is known that even a weak law of large numbers cannot hold. For the intermediate case \(p_n(m_n-k+1)=1\) the author derives a law of large numbers for certain weighted partial sums of the form \(\sum_{n=k}^Na_{nN}X_{n(k)}\) in the sense of complete convergence. A finite nonzero limit is obtained although the first moment of \(X_{n(k)}\) is infinite.