Difference between revisions of "M-dependent-process"
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− | There are one-dependent processes which are not | + | A discrete-time [[Stochastic process|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 [[#References|[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, [[#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 ![]() |
[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 ![]() |
[a4] | G. Haiman, "Valeurs extrémales de suites stationaires de variable aléatoires ![]() |
M-dependent-process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M-dependent-process&oldid=47740