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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" />.
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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
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<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>
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Let $( F , \mathcal{B} )$ be a [[Measurable space|measurable space]]. A sequence of random variables $X = ( X _ { n } ) _ { n \in Z }$ taking values in $F$ is described by the triple $( F ^ {\bf Z } , {\cal B} ^ {\bf Z } , \mathsf{P} )$, where $\textsf{P}$ is a [[Probability measure|probability measure]] on $( F ^ { \mathbf{Z} } , B ^ {\mathbf{Z} } )$ called the distribution of $X$. The sequence $X$ is said to be independent if $\textsf{P}$ is a product measure, i.e. $\mathsf{P} = \prod _ { x \in \mathbf{Z} } \mu _ { x }$ for probability measures $\mu _ { x }$ on $( F , \mathcal{B} )$.
  
<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>
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The right and left tail-sigma-fields of $X$ are defined as
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\begin{equation*} \mathcal{T} ^ { + } = \bigcap _ { N \geq 0 } \sigma ( X _ { n } : n \geq N ) \end{equation*}
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\begin{equation*} \mathcal{T} ^ { - } = \bigcap _ { N \geq 0 } \sigma ( X _ { n } : n \leq - N ) \end{equation*}
  
 
and the two-sided tail-sigma-field is defined as
 
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>
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\begin{equation*} \mathcal{T} = \bigcap _ { N \geq 0 } \sigma ( X _ { n } : | n | \geq N ). \end{equation*}
  
(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" .
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(Here, $\sigma ( Y )$ denotes the smallest sigma-field (cf. [[Borel field of sets|Borel field of sets]]) with respect to which $Y$ is measurable.) The Kolmogorov [[Zero-one law|zero-one law]] [[#References|[a1]]] states that, in the independent case, ${\cal T} ^ { + }$, $\mathcal{T}^{-}$ and $\mathcal{T}$ are trivial, i.e. all their elements have probability $0$ or $1$ under $\textsf{P}$. Without the independence property this need no longer be true: tail triviality only holds when $X$ has sufficiently weak dependencies. In fact, when the index set $\bf Z$ 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 ${\cal T} ^ { + }$, $\mathcal{T}^{-}$ are trivial but $\mathcal{T}$ 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]]].
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Instead of indexing the random variables by $\bf Z$ one may also consider a [[Random field|random field]] $( X _ { n } ) _ { n \in {\bf Z} ^ { d }}$, indexed by the $d$-dimensional integers ($d \geq 1$). The definition of $\mathcal{T}$ is the same as before, but now with $| n | = \operatorname { min } _ { 1 \leq i \leq d } | n _ { i } |$, and $\mathcal{T}$ is called the sigma-field at infinity. For independent random fields, $\mathcal{T}$ 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 $\textsf{P}$ is an extremal Gibbs measure [[#References|[a2]]].
  
 
====References====
 
====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>
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<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>

Latest revision as of 16:56, 1 July 2020

Let $( F , \mathcal{B} )$ be a measurable space. A sequence of random variables $X = ( X _ { n } ) _ { n \in Z }$ taking values in $F$ is described by the triple $( F ^ {\bf Z } , {\cal B} ^ {\bf Z } , \mathsf{P} )$, where $\textsf{P}$ is a probability measure on $( F ^ { \mathbf{Z} } , B ^ {\mathbf{Z} } )$ called the distribution of $X$. The sequence $X$ is said to be independent if $\textsf{P}$ is a product measure, i.e. $\mathsf{P} = \prod _ { x \in \mathbf{Z} } \mu _ { x }$ for probability measures $\mu _ { x }$ on $( F , \mathcal{B} )$.

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

\begin{equation*} \mathcal{T} ^ { + } = \bigcap _ { N \geq 0 } \sigma ( X _ { n } : n \geq N ) \end{equation*}

\begin{equation*} \mathcal{T} ^ { - } = \bigcap _ { N \geq 0 } \sigma ( X _ { n } : n \leq - N ) \end{equation*}

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

\begin{equation*} \mathcal{T} = \bigcap _ { N \geq 0 } \sigma ( X _ { n } : | n | \geq N ). \end{equation*}

(Here, $\sigma ( Y )$ denotes the smallest sigma-field (cf. Borel field of sets) with respect to which $Y$ is measurable.) The Kolmogorov zero-one law [a1] states that, in the independent case, ${\cal T} ^ { + }$, $\mathcal{T}^{-}$ and $\mathcal{T}$ are trivial, i.e. all their elements have probability $0$ or $1$ under $\textsf{P}$. Without the independence property this need no longer be true: tail triviality only holds when $X$ has sufficiently weak dependencies. In fact, when the index set $\bf Z$ 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 ${\cal T} ^ { + }$, $\mathcal{T}^{-}$ are trivial but $\mathcal{T}$ is not [a3]. Intuitively, in such examples there are "dependencies across infinity" .

Instead of indexing the random variables by $\bf Z$ one may also consider a random field $( X _ { n } ) _ { n \in {\bf Z} ^ { d }}$, indexed by the $d$-dimensional integers ($d \geq 1$). The definition of $\mathcal{T}$ is the same as before, but now with $| n | = \operatorname { min } _ { 1 \leq i \leq d } | n _ { i } |$, and $\mathcal{T}$ is called the sigma-field at infinity. For independent random fields, $\mathcal{T}$ 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 $\textsf{P}$ 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
How to Cite This Entry:
Tail triviality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tail_triviality&oldid=16478
This article was adapted from an original article by F. den Hollander (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article