We discuss issues of existence and stochastic modeling in regard to sequences that exhibit combined features of independence and instability of relative frequencies of marginal events. The concept of independence used here is borrowed from the frequentist account of numerical probability advanced by von Mises: A sequence is independent if certain salient asymptotic properties are invariant under the causal selection of subsequences. We show that independence (in the above sense) and instability of relative frequency are indeed compatible and that sequences with such features support stochastic models expressed in terms of envelopes of probability measures.
"Unstable Collectives and Envelopes of Probability Measures." Ann. Probab. 19 (2) 893 - 906, April, 1991. https://doi.org/10.1214/aop/1176990457