Certain mixing sequences of dependent `rare' events are considered and a Poisson limit is established for the probability that $k$ events occur. An asymptotic distribution for the number of upcrossings of a high level by certain stochastic processes is considered as an application.
"A Poisson-Type Limit Theorem for Mixing Sequences of Dependent 'Rare' Events." Ann. Probab. 1 (3) 480 - 483, June, 1973. https://doi.org/10.1214/aop/1176996941