We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A celebrated result of E. Lindenstrauss shows that normalized sums over certain increasing subsets of such groups approximate expectations. Our results clarify that the corresponding unnormalized sums of binary statistics are asymptotically Poisson, provided suitable mixing conditions hold. They extend further to randomly subsampled sums and also show that strict invariance of the distribution is not needed if the requisite mixing condition defined by the group holds. We illustrate the results with applications to random fields, Cayley graphs, and Poisson processes on groups.

翻译：我们将泊松极限定理推广至其分布具有可数群作用不变性的随机对象的二元函数。实例包括平稳随机场、可交换序列及可交换图。E. Lindenstrauss的著名结果表明，此类群上特定递增子集上的归一化和可逼近期望值。本文结果阐明，在适当混合条件下，相应的非归一化二元统计量之和渐近服从泊松分布。该结论进一步扩展至随机子采样和，并表明若由群定义的所需混合条件成立，则无需严格分布不变性。我们通过随机场、Cayley图及群上泊松过程的应用实例说明这些结果。