Tail triviality

Let be a measurable space. A sequence of random variables taking values in is described by the triple , where is a probability measure on called the distribution of . The sequence is said to be independent if is a product measure, i.e. for probability measures on .

The right and left tail-sigma-fields of are defined as

and the two-sided tail-sigma-field is defined as

(Here, denotes the smallest sigma-field (cf. Borel field of sets) with respect to which is measurable.) The Kolmogorov zero-one law [a1] states that, in the independent case, , and are trivial, i.e. all their elements have probability or under . Without the independence property this need no longer be true: tail triviality only holds when has sufficiently weak dependencies. In fact, when the index set is viewed as time, tail triviality means that the present is asymptotically independent of the far future and the far past. There exist examples where , are trivial but is not [a3]. Intuitively, in such examples there are "dependencies across infinity" .

Instead of indexing the random variables by one may also consider a random field , indexed by the -dimensional integers (). The definition of is the same as before, but now with , and is called the sigma-field at infinity. For independent random fields, is again trivial. Without the independence property, however, the question is considerably more subtle and is related to the phenomenon of phase transition (i.e. non-uniqueness of probability measures having prescribed conditional probabilities in finite sets). Tail triviality holds, for instance, when is an extremal Gibbs measure [a2].

References

[a1]	P. Billingsley, "Probability and measure" , Wiley (1986) (Edition: Second)
[a2]	H.-O. Georgii, "Gibbs measures and phase transitions" , Studies Math. , 9 , W. de Gruyter (1988)
[a3]	D.S. Ornstein, B. Weiss, "Every transformation is bilaterally deterministic" Israel J. Math. , 24 (1975) pp. 154–158