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In the distributed coding of correlated sources, the problem of characterizing the joint probability distribution of a pair of random variables satisfying an n-letter Markov chain arises. The exact solution of this problem is intractable. In this paper, we seek a single-letter necessary condition for this n-letter Markov chain. To this end, we propose a new data processing inequality on a new measure of correlation through a spectral method. Based on this new data processing inequality, we provide a single-letter necessary condition for the required joint probability distribution. We apply our results to two specific examples involving the distributed coding of correlated sources: multiple-access channel with correlated sources and multiterminal rate-distortion region, and propose new necessary conditions for these two problems.