TY - JOUR

T1 - A phase transition for the probability of being a maximum among random vectors with general iid coordinates

AU - Jacobovic, Royi

AU - Zuk, Or

N1 - Publisher Copyright:
© 2023 Elsevier B.V.

PY - 2023/8

Y1 - 2023/8

N2 - 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.

AB - 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.

KW - Extreme values

KW - Multivariate maximum

KW - Pareto

KW - Phase transition

UR - http://www.scopus.com/inward/record.url?scp=85153620801&partnerID=8YFLogxK

U2 - 10.1016/j.spl.2023.109847

DO - 10.1016/j.spl.2023.109847

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AN - SCOPUS:85153620801

SN - 0167-7152

VL - 199

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

M1 - 109847

ER -