Stochastic domination and weak convergence of conditioned Bernoulli random vectors

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.