The Annals of Probability

Stationary processes whose filtrations are standard

X. Bressaud, A. Maass, S. Martinez, and J. San Martin

Full-text: Open access


We study the standard property of the natural filtration associated to a 0–1 valued stationary process. In our main result we show that if the process has summable memory decay, then the associated filtration is standard. We prove it by coupling techniques. For a process whose associated filtration is standard, we construct a product type filtration extending it, based upon the usual couplings and the Vershik’s criterion for standardness.

Article information

Ann. Probab., Volume 34, Number 4 (2006), 1589-1600.

First available in Project Euclid: 19 September 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 60G10: Stationary processes

Standard filtrations summable memory decay couplings


Bressaud, X.; Maass, A.; Martinez, S.; San Martin, J. Stationary processes whose filtrations are standard. Ann. Probab. 34 (2006), no. 4, 1589--1600. doi:10.1214/009117906000000151.

Export citation


  • Bowen, R. (1975). Equilibrium States in Ergodic Theory of Anosov Diffeomorphism. Springer, Berlin.
  • Bressaud, X., Fernández, R. and Galves, A. (1999). Decay of correlations for non Hölderian dynamics. A coupling approach. Electron. J. Probab. 4 1--19.
  • Comets, F., Fernández, R. and Ferrari, P. (2002). Process with long memory, regenerative construction and perfect simulation. Ann. Appl. Probab. 12 921--943.
  • Dubins, L., Feldman, J., Smorodinsky, M. and Tsirelson, B. (1996). Decreasing sequences of $\sigma$-fields and a measure change for Brownian motion. Ann. Probab. 24 882--904.
  • Ferrari, P., Maass, A., Martínez, S. and Ney, P. (2000). Cesàro mean distribution of group automata staring from measures with summable decay. Ergodic Theory and Dynamical Systems 20 1657--1670.
  • Émery, M. (2002). Old and new tools in the theory of filtrations. Dynamics and Randomness. Coll. Nonlinear Phen. Compl. Syst. 7 125--146.
  • Émery, M. and Schachermayer, W. (2001). On Vershik's standardness criterion and Tsirelson's notion of cosiness. Séminaire de Probabilités XXXV. Lecture Notes in Math. 1755 265--305. Springer, Berlin.
  • Feldman, J. and Smorodinsky, M. (2000). Decreasing sequences of measurable partitions: Product type, standard, and prestandard. Ergodic Theory and Dynamical Systems 20 1079--1090.
  • Lalley, S. (1986). Regenerative representation for one-dimensional Gibbs states. Ann. Probab. 14 1262--1271.
  • Laurent, S. (2004). Filtrations à temps discret négatif. Thèse de doctorat, U. Strasbourg. Available at
  • Masani, P. (1966). Wiener's contribution to generalized harmonic analysis, prediction theory and filter theory. Bull. Amer. Math. Soc. 72 73--125.
  • Rosenblatt, M. (1959). Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8 665--681.
  • Ruelle, D. (1978). Thermodynamic Formalism. Addsison--Wesley, Reading, MA.
  • Tsirelson, B. (1997). Triple points: From non-Brownian filtrations to harmonic measures. Geom. Funct. Anal. 7 1096--1142.
  • Vershik, A. M. (1973). Approximation in measure theory. Dissertation, Leningrad Univ. In Russian.
  • Vershik, A. M. (1995). The theory of decreasing sequences of measurable partitions. St. Petersburg Math. J. 6 705--761.