### Abstract

Let X_{1}, X_{2}, ..., 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.

Original language | English (US) |
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Pages (from-to) | 226-241 |

Number of pages | 16 |

Journal | Journal of Applied Probability |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2003 |

Externally published | Yes |

### Keywords

- Asymptotics
- Clustering
- Extremes
- Law of large numbers
- Order statistics

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty