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

翻译：我们考虑一个由美元独立的二进制变量构成的矢量,每个变量的成功概率不同。以其总和为条件的矢量分布被称为有条件的伯努利分布。假设美元是无穷无尽的,而且这一数额与美元成正比,精确的取样成本为2美元,而一个简单的Markov连锁的蒙特卡洛算法使用“擦拭”每迭代都有固定的成本。我们提供了马可夫链条汇合的条件,以美元为Nlog NN美元迭代值。我们的证据依赖于将空间分割成有利和不受欢迎的对子的组合和辅助马可夫链。