Pointwise inequalities for ergodic averages and reversed martingales

It is known that the ergodic averages Anϕ in the context of the shift action on Z satisfy pointwise inequalities of the form Anϕ C E(ϕ|Fn)+ E(ϕ|Gn) , where {Fn}n 1 and {Gn}n 1 are decreasing sequences of σ -algebras on Z. In this paper we extend this by examining situations when the ergodic averages can be pointwise dominated by one reversed martingale, and situations when a reversed martingale can be pointwise dominated by ergodic averages.  2003 Elsevier Inc. All rights reserved.