On Modified Logarithmic Sobolev Inequalities for Bernoulli and Poisson Measures

We show that for any positive functionfon the discrete cube {0, 1}n,Entμnp(f)⩽pq Eμnp 1f |Df|2whereμnpis the product measure of the Bernoulli measure with probability of successp, as well as related inequalities, which may be shown to imply in the limit the classical Gaussian logarithmic Sobolev inequality as well as a logarithmic Sobolev inequality for Poisson measure. We further investigate modified logarithmic Sobolev inequalities to analyze integrability properties of Lipschitz functions on discrete spaces. In particular, we obtain, under modified logarithmic Sobolev inequalities, some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables.