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:
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Received: 6 May 1996 / In revised form: 29 September 1997
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Marton, K. Measure concentration for a class of random processes. Probab Theory Relat Fields 110, 427–439 (1998). https://doi.org/10.1007/s004400050154
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DOI: https://doi.org/10.1007/s004400050154