DeepAI AI Chat
Log In Sign Up

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

by   Royi Jacobovic, et al.
Hebrew University of Jerusalem

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→∞.


page 1

page 2

page 3

page 4


Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions

The Gaussian product inequality is an important conjecture concerning th...

The level crossings of random sums

Let {a_j}_j = 0^N and {b_j}_j = 0^N be sequences of mutually independent...

Computing Generalized Convolutions Faster Than Brute Force

In this paper, we consider a general notion of convolution. Let D be a f...

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

Consider n iid real-valued random vectors of size k having iid coordinat...

A Statistical Approach to Broken Stick Problems

Let a stick be broken at random at n-1 points to form n pieces. We consi...

Estimating the probability that a given vector is in the convex hull of a random sample

For a d-dimensional random vector X, let p_n, X be the probability that ...

Fitting a manifold of large reach to noisy data

Let M⊂R^n be a C^2-smooth compact submanifold of dimension d. Assume tha...