M-dependent-process

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