Stochastic domination and weak convergence of conditioned Bernoulli random vectors

Journal article, 2012

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) over tilde (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) over tilde (n) similarly, but with success probabilities q(i) >= p(i). We prove that the law of (X) over tilde (n) converges weakly to a distribution that we can describe precisely. We then prove that sup P((X) over tilde <= (Y) over tilde (n)) converges to a constant, where the supremum is taken over all possible couplings of (X) over tilde and (Y) over tilde (n). This constant is expressed explicitly in terms of the parameters of the system.