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

2021/12/31 by Jacobovic, Royi, Zuk, Or
#60C05 #60F20 #62E20 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2112.15534

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 which 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 is growing 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.

Related

• Implicit Extremes and Implicit Max-Stable Laws
• An Offspring of Multivariate Extreme-Value Theory: The\n Max-Characteristic Function
• On generalized max-linear models and their statistical interpolation
• On the Maximum of Random Variables on Product Spaces
• A note on the maximum of the Riemann zeta function, and log-correlated random variables