We study tail a-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. 1. Statement of results. Let X 1 , X 2 , ••• be a strictly stationary sequence of integer-valued random variables and let S 1 , 8 2 , • • • be the sums n;;:;:: 1. If the Xi are independent, then according to the Hewitt-Savage zero-one law [Breiman (1968) ] the tail a-field is trivial. Of

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... se, without the independence this need no longer be true. The main question that will be addressed in this paper is what can be said about the tail behavior of the sums in this more general setting. An early reference is Blackwell and Freedman (1964), where the Hewitt-Savage zero-one law is generalized to Markov sequences. A later reference is Georgii (1976, ~.1979) for Gibbs states with finite state space. It is natural to extend (Xn)n~I to a double-sided process X == (Xn)nez and to extend also the sums to a double-sided process (Sn)nez by requiring that (1.1) So= 0, The process X is defined on the probability space (0, .ff', P) with n = Z z, X the identity on n, .ff' the product a-field generated by the discrete topology on z and P a T-invariant probability measure with T the shift defined by (TX)n = Xn+ 1 , n E Z. We shall be interested in the following tail a-fields associated with