Stochastic domination and weak convergence of conditioned Bernoulli random vectors

TL;DR

This paper investigates the weak convergence and stochastic domination of conditioned Bernoulli vectors with block structures, providing explicit descriptions of limiting distributions and bounds on coupling probabilities.

Contribution

It establishes the weak convergence of conditioned Bernoulli vectors and explicitly characterizes the maximal coupling probability between two such vectors.

Findings

Weak convergence of conditioned Bernoulli vectors is proven.

Explicit formulas for the limiting distribution are derived.

The supremum of coupling probabilities converges to a constant expressed in system parameters.

Abstract

For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M "blocks" such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size of each block is essentially linear in n. Let X'_n be a random vector having the conditional distribution of X_n, conditioned on the total number of successes being at least k_n, where k_n is also essentially linear in n. Define Y'_n similarly, but with success probabilities q_i>=p_i. We prove that the law of X'_n converges weakly to a distribution that we can describe precisely. We then prove that sup Pr(X'_n <= Y'_n) converges to a constant, where the supremum is taken over all possible couplings of X'_n and Y'_n. This constant is expressed explicitly in terms of the parameters of the system.