Mathematics – Probability
Scientific paper
2011-09-27
Mathematics
Probability
39 pages, 2 figures
Scientific paper
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.
Broman Erik
de Brug Tim van
Kager Wouter
Meester Ronald
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