Difference between revisions of "M-dependent-process"
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A discrete-time [[Stochastic process|stochastic process]] $ ( X _ {n} ) _ {n \in \mathbf Z } $ | A discrete-time [[Stochastic process|stochastic process]] $ ( X _ {n} ) _ {n \in \mathbf Z } $ | ||
| − | is $ m $- | + | is $ m $-dependent if for all $ k $ |
| − | dependent if for all $ k $ | ||
the joint stochastic variables $ ( X _ {n} ) _ {n \leq k } $ | the joint stochastic variables $ ( X _ {n} ) _ {n \leq k } $ | ||
are independent of the joint stochastic variables $ ( X _ {n} ) _ {n \geq k + m + 1 } $. | 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 $- | + | 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 } $ |
| − | 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 $ | be an independent process and $ \phi $ | ||
a function of $ m + 1 $ | a function of $ m + 1 $ | ||
| − | variables; let $ X _ {n} = f ( Z _ {n} \dots Z _ {n+} | + | variables; let $ X _ {n} = f ( Z _ {n} \dots Z _ {n+m} ) $; |
| − | then the $ ( m + 1 ) $- | + | then the $ ( m + 1 ) $-block factor $ X _ {n} $ |
| − | block factor $ X _ {n} $ | + | is an $ m $-dependent process. |
| − | is an $ m $- | ||
| − | dependent process. | ||
| − | There are one-dependent processes which are not $ 2 $- | + | There are one-dependent processes which are not $ 2 $-block factors, [[#References|[a2]]]. |
| − | block factors, [[#References|[a2]]]. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> G.L. O'Brien, "Scaling transformations for $\{ 0,1 \}$-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 $m$-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 $m$-dépendantes" ''Ann. Inst. H. Poincaré Sect. B (N.S.)'' , '''17''' (1981) pp. 309–330</td></tr></table> |
Latest revision as of 12:34, 18 February 2022
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 $\{ 0,1 \}$-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 $m$-dependent sequences" Ann. Probab. , 12 (1984) pp. 805–818 |
| [a4] | G. Haiman, "Valeurs extrémales de suites stationaires de variable aléatoires $m$-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
M-dependent-process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M-dependent-process&oldid=47740