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A discrete-time [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m0620002.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m0620004.png" />-dependent if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m0620005.png" /> the joint stochastic variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m0620006.png" /> are independent of the joint stochastic variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m0620007.png" />.
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Such processes arise naturally as limits of rescaling transformations (renormalizations) and (hence) as examples of processes with scaling symmetries [[#References|[a1]]]. Examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m0620008.png" />-dependent processes are given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200010.png" />-block factors. These are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200011.png" /> be an independent process and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200012.png" /> a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200013.png" /> variables; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200014.png" />; then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200015.png" />-block factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200016.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200017.png" />-dependent process.
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There are one-dependent processes which are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200018.png" />-block factors, [[#References|[a2]]].
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A discrete-time [[Stochastic process|stochastic process]]  $  ( X _ {n} ) _ {n \in \mathbf Z }  $
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is  $  m $-
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dependent if for all  $  k $
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the joint stochastic variables  $  ( X _ {n} ) _ {n \leq  k }  $
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are independent of the joint stochastic variables  $  ( X _ {n} ) _ {n \geq  k + m + 1 }  $.
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Such processes arise naturally as limits of rescaling transformations (renormalizations) and (hence) as examples of processes with scaling symmetries [[#References|[a1]]]. Examples of  $  m $-
 +
dependent processes are given by  $  ( m + 1 ) $-
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block factors. These are defined as follows. Let  $  ( Z _ {n} ) _ {n \in \mathbf Z }  $
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be an independent process and  $  \phi $
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a function of  $  m + 1 $
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variables; let  $  X _ {n} = f ( Z _ {n} \dots Z _ {n+} m ) $;
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then the  $  ( m + 1 ) $-
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block factor  $  X _ {n} $
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is an  $  m $-
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dependent process.
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There are one-dependent processes which are not $  2 $-
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block factors, [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.L. O'Brien,  "Scaling transformations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200019.png" />-valued sequences"  ''Z. Wahrscheinlichkeitstheorie Verw. Gebiete'' , '''53'''  (1980)  pp. 35–49</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Aaronson,  D. Gilat,  M. Keane,  V. de Valk,  "An algebraic construction of a class of one-dependent processes"  ''Ann. Probab.'' , '''17'''  (1988)  pp. 128–143</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Janson,  "Runs in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200020.png" />-dependent sequences"  ''Ann. Probab.'' , '''12'''  (1984)  pp. 805–818</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Haiman,  "Valeurs extrémales de suites stationaires de variable aléatoires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200021.png" />-dépendantes"  ''Ann. Inst. H. Poincaré Sect. B (N.S.)'' , '''17'''  (1981)  pp. 309–330</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.L. O'Brien,  "Scaling transformations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200019.png" />-valued sequences"  ''Z. Wahrscheinlichkeitstheorie Verw. Gebiete'' , '''53'''  (1980)  pp. 35–49</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Aaronson,  D. Gilat,  M. Keane,  V. de Valk,  "An algebraic construction of a class of one-dependent processes"  ''Ann. Probab.'' , '''17'''  (1988)  pp. 128–143</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Janson,  "Runs in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200020.png" />-dependent sequences"  ''Ann. Probab.'' , '''12'''  (1984)  pp. 805–818</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Haiman,  "Valeurs extrémales de suites stationaires de variable aléatoires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062000/m06200021.png" />-dépendantes"  ''Ann. Inst. H. Poincaré Sect. B (N.S.)'' , '''17'''  (1981)  pp. 309–330</TD></TR></table>

Revision as of 04:11, 6 June 2020


A discrete-time stochastic process $ ( X _ {n} ) _ {n \in \mathbf Z } $ is $ m $- dependent if for all $ k $ the joint stochastic variables $ ( X _ {n} ) _ {n \leq k } $ are independent of the joint stochastic variables $ ( X _ {n} ) _ {n \geq k + m + 1 } $.

Such processes arise naturally as limits of rescaling transformations (renormalizations) and (hence) as examples of processes with scaling symmetries [a1]. Examples of $ m $- dependent processes are given by $ ( m + 1 ) $- block factors. These are defined as follows. Let $ ( Z _ {n} ) _ {n \in \mathbf Z } $ be an independent process and $ \phi $ a function of $ m + 1 $ variables; let $ X _ {n} = f ( Z _ {n} \dots Z _ {n+} m ) $; then the $ ( m + 1 ) $- block factor $ X _ {n} $ is an $ m $- dependent process.

There are one-dependent processes which are not $ 2 $- block factors, [a2].

References

[a1] G.L. O'Brien, "Scaling transformations for -valued sequences" Z. Wahrscheinlichkeitstheorie Verw. Gebiete , 53 (1980) pp. 35–49
[a2] J. Aaronson, D. Gilat, M. Keane, V. de Valk, "An algebraic construction of a class of one-dependent processes" Ann. Probab. , 17 (1988) pp. 128–143
[a3] S. Janson, "Runs in -dependent sequences" Ann. Probab. , 12 (1984) pp. 805–818
[a4] G. Haiman, "Valeurs extrémales de suites stationaires de variable aléatoires -dépendantes" Ann. Inst. H. Poincaré Sect. B (N.S.) , 17 (1981) pp. 309–330
How to Cite This Entry:
M-dependent-process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M-dependent-process&oldid=47740