## Abstract

Consider n iid real-valued random vectors of size k having iid coordinates with a general distribution function F. A vector is a maximum if and only if there is no other vector in the sample that weakly dominates it in all coordinates. Let p_{k,n} be the probability that the first vector is a maximum. The main result of the present paper is that if k≡k_{n} grows at a slower (faster) rate than a certain factor of log(n), then p_{k,n}→0 (resp. p_{k,n}→1) as n→∞. Furthermore, the factor is fully characterized as a functional of F. We also study the effect of F on p_{k,n}, showing that while p_{k,n} may be highly affected by the choice of F, the phase transition is the same for all distribution functions up to a constant factor.

Original language | American English |
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Article number | 109847 |

Journal | Statistics and Probability Letters |

Volume | 199 |

DOIs | |

State | Published - Aug 2023 |

### Bibliographical note

Publisher Copyright:© 2023 Elsevier B.V.

## Keywords

- Extreme values
- Multivariate maximum
- Pareto
- Phase transition