Measure concentration for a class of random processes

Abstract

Let X = {Xi}∞i=-∞ be a stationary random process with a countable alphabet X and distribution q. Let q∞ (·|x0-k) denote the conditional distribution of X∞ = (X1, X2, . . . , Xn, . . .) given the k-length past: q∞ (·|x0-k) = dist (X∞|X0-k = x0-k) . Write d(x̂1, x1) = 0 if x̂1 = x1, and d(x̂1, x̂1) = 1 otherwise. We say that the process X admits a joining with finite distance u if for any two past sequences x̂0-k = (x̂-k+1, . . . , x̂0) and x̂0-k = (x-k+1, . . . , x0), there is a joining of q∞ (·|x0-k) and q∞ (·|x0-k), say dist(X̂∞0, X∞0|x̂0-k, x0-k), such that E{∑∞i=1 d(X̂i, Xi)|x̂0-k, x0-k} ≤ u .