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 a coherent upper prevision. 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：严格频率主义将概率定义为无限序列中相对频率的极限。若该极限不存在会怎样？我们提出一种更广泛的理论，同样适用于表现出发散相对频率的随机现象。在此过程中，我们发展了与非精确概率理论的紧密联系：相对频率的聚点构成一个一致的上预期望。我们证明，条件概率的自然频率主义定义可恢复广义贝叶斯规则。这也暗示了一个独立性概念，该概念与非精确概率文献中的认知无关性相关。最后，我们建设性地证明，对于有限基本事件集，存在一个序列，其相对频率的聚点与预设集合一致，这显示了本理论的自然性，并可谓完备性。