Summary: Ann. Inst. H. Poincaré, Probab. Statist. 37, 1 (2001) 101­137
© 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved
S0246-0203(00)01061-X/FLA
CLARK­OCONE FORMULAS AND POINCARÉ
INEQUALITIES ON THE DISCRETE CUBE
Cécile ANÉ
Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118, route de Narbonne,
31062 Toulouse, France
Received 12 January 2000, revised 17 April 2000
ABSTRACT. ­ We establish Poincaré inequalities for the continuous time random walk on the
cube {-1,+1}d. A first method is based on the study of cylindrical functionals. A Poincaré
inequality is proved for these functionals and extended to arbitrary functionals. A second method
is based on martingale representation formulas. A whole family of Clark­Ocone formulas is
then available, which leads to the corresponding family of Poincaré inequalities. These various
inequalities are compared through examples. We also show that the cylindrical method extends
to some asymmetric continous time random walks on {-1,+1}d. © 2001 Éditions scientifiques
et médicales Elsevier SAS
AMS classification: 60H07, 60J27
RÉSUMÉ. ­ Nous établissons des inégalités de Poincaré pour la marche aléatoire à temps
continu sur le cube {-1,+1}d. Une première méthode consiste à étudier en premier lieu les

Source: Ané, Cécile - Departments of Botany & Statistics, University of Wisconsin at Madison

Collections: Biology and Medicine; Mathematics