Exact laws of order statistics for φ-mixing arrays of Pareto random variables

For m≥1, let {Xnj,1≤j≤m,n≥1} be a φ-mixing array of Pareto random variables. Denote Xn(k) as the k-th order statistic from the n-th row and define the ratio order statistic Rnij=Xn(j)/Xn(i) for 1≤i<j≤m. This study establishes exact weak laws and strong laws of large numbers for sums of the k-th order statistic Xn(k) and ratio order statistic Rnij, even in cases where EXn(k) and ERnij do not exist. Notably, we demonstrate that these exact weak laws cannot be extended to strong laws, revealing a fundamental limitation in the convergence behavior of such statistics. Furthermore, to illustrate the derived exact laws, we conduct simulations by generating φ-mixing sequences based on Pareto random variables.