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We consider a vector of $N$ independent binary variables, each with adifferent probability of success. The distribution of the vector conditional onits sum is known as the conditional Bernoulli distribution. Assuming that $N$goes to infinity and that the sum is proportional to $N$, exact sampling costsorder $N^2$, while a simple Markov chain Monte Carlo algorithm using 'swaps'has constant cost per iteration. We provide conditions under which this Markovchain converges in order $N \log N$ iterations. Our proof relies on couplingsand an auxiliary Markov chain defined on a partition of the space intofavorable and unfavorable pairs.

A simple Markov chain for independent Bernoulli variables conditioned on their sum