A phase transition for the probability of being a maximum among random vectors with general iid coordinates

Royi Jacobovic*, Or Zuk

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 pk,n be the probability that the first vector is a maximum. The main result of the present paper is that if k≡kn grows at a slower (faster) rate than a certain factor of log(n), then pk,n→0 (resp. pk,n→1) as n→∞. Furthermore, the factor is fully characterized as a functional of F. We also study the effect of F on pk,n, showing that while pk,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 languageAmerican English
Article number109847
JournalStatistics and Probability Letters
Volume199
DOIs
StatePublished - Aug 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier B.V.

Keywords

  • Extreme values
  • Multivariate maximum
  • Pareto
  • Phase transition

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