We study tail $\sigma$-fields and loss of memory associated with sums of stationary integer-valued random variables. An application concerns convergence in distribution of interarrival times in zero-one sequences.
H. C. P. Berbee. W. Th. F. Den Hollander. "Tail Triviality for Sums of Stationary Random Variables." Ann. Probab. 17 (4) 1635 - 1645, October, 1989. https://doi.org/10.1214/aop/1176991177