Uniqueness in g-measures

Abstract

Using coupling techniques extending ideas from Harris (1955 Pacific J. Math. 5 707-24), we prove uniqueness in g-measures and give estimates of the rates of convergence for the associated Markov chains, for strictly positive continuous g-functions under a weak regularity condition. Our regularity condition is weaker than the earlier weakest known conditions for uniqueness (Harris T E 1955 Pacific J. Math. 5 707-24; Iosifescu M and Spǎtaru A 1973 Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 195-214; Comets F et al 2002 Ann. Appl. Probab. 12 921-43). As a consequence of our method, we obtain sharper bounds on the rates of convergence also in cases when more restrictive regularity conditions are satisfied, and thus in particular, we extend results by Bressaud et al (1999 Electron. J. Probab. 4 19).