2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension

Thierry de La Rue (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process indexed by Z, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold mixing problem, the so-called "3-dot system", but in the context of stationary random fields indexed by ZxZ. In this work, we first present an attempt to adapt Ledrappier's construction to the one-dimensional case, which finally leads to a stationary process which is 2-fold but not 3-fold mixing conditionally to the sigma-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample can not be fully adapted to one-dimensional stationary processes.

Mots clés: multifold mixing; 3-dot system; self-joining

Domaines: Probabilités [math.PR]

Thierry de La Rue. 2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension. Bulletin of the Brazilian Mathematical Society, 2006, 37 (4), pp.503-521. ⟨hal-00020175⟩