  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.
