Strict frequentism defines probability as the limiting relative frequency in an infinite sequence. What if the limit does not exist? We present a broader theory, which is applicable also to random phenomena that exhibit diverging relative frequencies. In doing so, we develop a close connection with the theory of imprecise probability: the cluster points of relative frequencies yield an upper probability. We show that a natural frequentist definition of conditional probability recovers the generalized Bayes rule. This also suggests an independence concept, which is related to epistemic irrelevance in the imprecise probability literature. Finally, we prove constructively that, for a finite set of elementary events, there exists a sequence for which the cluster points of relative frequencies coincide with a prespecified set which demonstrates the naturalness, and arguably completeness, of our theory.

翻译：严格频率学派将概率定义为无限序列中相对频率的极限。如果该极限不存在怎么办？我们提出了一种更广泛的理论，同样适用于表现出发散相对频率的随机现象。在此过程中，我们建立了与不精确概率理论的紧密联系：相对频率的聚点产生一个上概率。我们证明了条件概率的自然频率学派定义能够恢复广义贝叶斯规则。这也暗示了一个独立概念，与不精确概率文献中的认识论无关性相关。最后，我们构造性地证明，对于有限基本事件集，存在一个序列，其相对频率的聚点与一个预先指定的集合一致，这展示了我们理论的自然性，甚至可以说是完备性。