A correlation inequality for random points in a hypercube with some implications

2022/09/01 by Jacobovic, Royi, Zuk, Or
#06-08 #06A07 #58E17 #60E15 #60F05 #62E20 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2209.00346

Abstract

Let \prec be the product order on ℝ^k and assume that X₁,X₂,…,X_n (n≥3) are i.i.d. random vectors distributed uniformly in the unit hypercube [0,1]^k. Let S be the (random) set of vectors in ℝ^k that \prec-dominate all vectors in \X₃,..,X_n\, and let W be the set of vectors that are not \prec-dominated by any vector in \X₃,..,X_n\. The main result of this work is the correlation inequality P(X₂∈ W|X₁∈ W)≤ P(X₂∈ W|X₁∈ S) . For every 1≤ i ≤ n let E_i,n be the event that X_i is not \prec-dominated by any of the other vectors in \X₁,…,X_n\. The main inequality yields an elementary proof for the result that the events E_1,n and E_2,n are asymptotically independent as n→∞. Furthermore, we derive a related combinatorial formula for the variance of the sum ∑_i=1ⁿ 1_E_i,n, i.e. the number of maxima under the product order \prec, and show that certain linear functionals of partial sums of \1_E_i,n;1≤ i≤ n\ are asymptotically normal as n→∞.

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