Abstract
We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μn on the discrete cube {0, 1}n and on the lattice Z n. In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z n. The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes.
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Received: 30 April 1997 / Revised version: 5 June 1998
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Bobkov, S., Götze, F. Discrete isoperimetric and Poincaré-type inequalities. Probab Theory Relat Fields 114, 245–277 (1999). https://doi.org/10.1007/s004400050225
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DOI: https://doi.org/10.1007/s004400050225
- Mathematics Subject Classification (1991): Primary 60E15; Secondary 26D15