Examples and counterexamples to almost-sure convergence of bilateral martingales

Abstract:

Given a stationary process $(X_p)_{p\in\ZZ}$ and an event $B\in \sigma(X_p,\, p\in\ZZ)$, we study the almost sure convergence as $n$ and $m$ go to infinity of the ``bilateral'' martingale $$ \espc{\ind{B}}{X_{-n}, X_{-n+1},\ldots,X_{m-1},X_m}. $$ We show that almost sure convergence holds in some classical examples such as i.i.d.~or Markov processes, as well as for the natural generator of Chacon's transformation. However, we also prove that in every aperiodic dynamical system with finite entropy, there exists a generating process and a measurable set $B$ for which the almost sure convergence of the bilateral martingale does not hold.

Laboratoire de Math\'ematiques Rapha\"el Salem, UMR 6085 CNRS--Universit\'e de Rouen, Site Colbert, F76821 Mont-Saint-Aignan Cedex, France
Thierry.Delarue@univ-rouen.fr
http://www.univ-rouen.fr/LMRS/Persopage/Delarue/index.html