Measure concentration for a class of random processes

Summary.   Let X={X _i } _{i =−∞} ^∞ be a stationary random process with a countable alphabet and distribution q. Let q ^∞(·|x _{− k} ⁰) denote the conditional distribution of X ^∞=(X ₁,X ₂,…,X _n ,…) given the k-length past: Write d(₁,x ₁)=0 if ₁=x ₁, and d(₁,x ₁)=1 otherwise. We say that the process X admits a joining with finite distance u if for any two past sequences _{− k} ⁰=(_{− k +1},…,₀) and x _{− k} ⁰=(x _{− k +1},…,x ₀), there is a joining of q ^∞(·|_{− k} ⁰) and q ^∞(·|x _{− k} ⁰), say dist(₀ ^∞,X ₀ ^∞|_{− k} ⁰,x _{− k} ⁰), such that The main result of this paper is the following inequality for processes that admit a joining with finite distance: Received: 6 May 1996 / In revised form: 29 September 1997