A clustering law for some discrete order statistics

Let X₁, X₂, ..., X_n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a 'clustering' condition, lim_{n → ∞}[1 -F(n + 1)]/[1 - F(n)] = 0. In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if lim_{n → ∞} [1 - F(n + r + 1)]/[1 - F(n)] = 0. Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.