A characterization of the strong law of large numbers for Bernoulli sequences

The law of large numbers is one of the most fundamental results in Probability Theory. In the case of independent sequences, there are some known characterizations; for instance, in the independent and identically distributed setting it is known that the law of large numbers is equivalent to integrability. In the case of dependent sequences, there are no known general characterizations – to the best of our knowledge. We provide such a characterization for identically distributed Bernoulli sequences in terms of a product disintegration.