Unusual strong laws for arrays of ratios of order statistics

Abstract

Let {X_n, k, 1 ≤ k ≤ m_n, n ≥ 1} be independent random variables from the Pareto distribution. Let X_n(k) be the kth largest order statistic from the nth row of our array, where X_n(1) denotes the largest order statistic from the nth row. Then set R_{n, in, jn} = X_n(jn) / X_n(in) where j_n < i_n. This paper establishes limit theorems involving weighted sums from the sequence {R_{n, in, jn}, n ≥ 1}, where for the first time we allow j_n → ∞, but only at a slow rate.