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
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and the two-sided tail-sigma-field is defined as
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(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 |
Tail triviality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tail_triviality&oldid=50128