Abstract
The concentration of measure phenomenon in product spaces means the following: if a subsetA of then'th power of a probability space Χ does not have too small a probability then very large probability is concentrated in a small neighborhood ofA. The neighborhood is in many cases understood in the sense of Hamming distance, and then measure concentration is known to occur for product probability measures, and also for the distribution of some processes with very fast and uniform decay of memory. Recently Talagrand introduced another notion of neighborhood of sets for which he proved a similar measure concentration inequality which in many cases allows more efficient applications than the one for a Hamming neighborhood. So far this inequality has only been proved for product distributions. The aim of this paper is to give a new proof of Talagrand's inequality, which admits an extension to contracting Markov chains. The proof is based on a new asymmetric notion of distance between probability measures, and bounding this distance by informational divergence. As an application, we analyze the bin packing problem for Markov chains.
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This work was supported by the grants OTKA 1906 and T 016386 of the Hungarian Academy of Sciences, and by MTA-NSF project 37.
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Marton, K. A measure concentration inequality for contracting markov chains. Geometric and Functional Analysis 6, 556–571 (1996). https://doi.org/10.1007/BF02249263
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DOI: https://doi.org/10.1007/BF02249263