View source for Tail triviality

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200201.png" /> be a [[Measurable space|measurable space]]. A sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200202.png" /> taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200203.png" /> is described by the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200204.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200205.png" /> is a [[Probability measure|probability measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200206.png" /> called the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200207.png" />. The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200208.png" /> is said to be independent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t1200209.png" /> is a product measure, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002010.png" /> for probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002012.png" />.

The right and left tail-sigma-fields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002013.png" /> are defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002014.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002015.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002016.png" /></td> </tr></table>

(Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002017.png" /> denotes the smallest sigma-field (cf. [[Borel field of sets|Borel field of sets]]) with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002018.png" /> is measurable.) The Kolmogorov [[Zero-one law|zero-one law]] [[#References|[a1]]] states that, in the independent case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002021.png" /> are trivial, i.e. all their elements have probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002022.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002023.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002024.png" />. Without the independence property this need no longer be true: tail triviality only holds when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002025.png" /> has sufficiently weak dependencies. In fact, when the index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002026.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002028.png" /> are trivial but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002029.png" /> is not [[#References|[a3]]]. Intuitively, in such examples there are  "dependencies across infinity" .

Instead of indexing the random variables by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002030.png" /> one may also consider a [[Random field|random field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002031.png" />, indexed by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002032.png" />-dimensional integers (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002033.png" />). The definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002034.png" /> is the same as before, but now with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002035.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002036.png" /> is called the sigma-field at infinity. For independent random fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002037.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120020/t12002038.png" /> is an extremal Gibbs measure [[#References|[a2]]].

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