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On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. (English) Zbl 0920.60002
Let \(\mu_{p}^{n}\) be the product measure on \(\{0,1\}^{n}\) of the Bernoulli measure with probability of success \(p\in [0,1]\) and let \(f\) be a positive function on \(\{0,1\}^{n}\). The authors prove first the following inequality: \[ \text{Ent}_{\mu_{p}^{n}}(f)\leq p(1-p)\text{E}_{\mu_{p}^{n}}\bigl(\tfrac{1}{f}| Df| ^{2}\bigr), \] where \(| Df| ^{2}(x)=\sum_{i=1}^{n}| f(x+e_{i})-f(x)| ^{2}\), \(x\in\{0,1\}^{n}\), \((e_{1}\ldots, e_{n})\) being the canonical basis of \(R^{n}\), and the addition is modulo 2. Then some related logarithmic Sobolev inequalities for Bernoulli and Poisson measures that will lead to some sharp form of modified logarithmic Sobolev inequalities are investigated [see also S. G. Bobkov and M. Ledoux, Probab. Theory Relat. Fields 107, No. 3, 383-400 (1997; Zbl 0878.60014)]. This type of inequalities entails some information on the Poisson behaviour of Lipschitz functions on discrete spaces. One obtains some concentration results for product measures [see also M. Ledoux, J. Math. Kyoto Univ. 35, No. 2, 211-220 (1995; Zbl 0836.60074) or ESAIM, Probab. Stat. 1, 63-87 (1997; Zbl 0869.60013) and also M. Talagrand, Publ. Math., Inst. Hautes Étud. Sci. 81, 73-205 (1995; Zbl 0864.60013)].

MSC:
60E15 Inequalities; stochastic orderings
28A35 Measures and integrals in product spaces
60E99 Distribution theory
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