  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.
