View source for M-dependent-process

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" />.

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.

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]]].

====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>