### 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) |
---|---|

Pages (from-to) | 226-241 |

Number of pages | 16 |

Journal | Journal of Applied Probability |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

**A clustering law for some discrete order statistics.** / Sethuraman, Sunder.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 40, no. 1, pp. 226-241. https://doi.org/10.1239/jap/1044476836

}

TY - JOUR

T1 - A clustering law for some discrete order statistics

AU - Sethuraman, Sunder

PY - 2003/3

Y1 - 2003/3

N2 - Let X1, X2, ..., Xn 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, limn → ∞[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 limn → ∞ [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.

AB - Let X1, X2, ..., Xn 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, limn → ∞[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 limn → ∞ [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.

KW - Asymptotics

KW - Clustering

KW - Extremes

KW - Law of large numbers

KW - Order statistics

UR - http://www.scopus.com/inward/record.url?scp=0037573234&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037573234&partnerID=8YFLogxK

U2 - 10.1239/jap/1044476836

DO - 10.1239/jap/1044476836

M3 - Article

AN - SCOPUS:0037573234

VL - 40

SP - 226

EP - 241

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 1

ER -