A correlation inequality for random points in a hypercube with some implications
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Let ≺ be the product order on ℝ^k and assume that X_1,X_2,…,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 ≺-dominate all vectors in {X_3,..,X_n}, and let W be the set of vectors that are not ≺-dominated by any vector in {X_3,..,X_n}. The main result of this work is the correlation inequality P(X_2∈ W|X_1∈ W)≤ P(X_2∈ W|X_1∈ S) . For every 1≤ i ≤ n let E_i,n be the event that X_i is not ≺-dominated by any of the other vectors in {X_1,…,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=1^n 1_E_i,n, i.e. the number of maxima under the product order ≺, 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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