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M-dependent-process

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A discrete-time stochastic process is -dependent if for all the joint stochastic variables are independent of the joint stochastic variables .

Such processes arise naturally as limits of rescaling transformations (renormalizations) and (hence) as examples of processes with scaling symmetries [a1]. Examples of -dependent processes are given by -block factors. These are defined as follows. Let be an independent process and a function of variables; let ; then the -block factor is an -dependent process.

There are one-dependent processes which are not -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://www.encyclopediaofmath.org/index.php?title=M-dependent-process&oldid=14932