Functional inequalities for discrete gradients and application to the geometric distribution

Abstract

Abstract. We present several functional inequalities for finite difference gradients, such as a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities, associated deviation estimates, and an exponential integrability property. In the particular case of the geometric distribution on N we use an integration by parts formula to compute the optimal isoperimetric and Poincaré constants, and to obtain an improvement of our general logarithmic Sobolev inequality. By a limiting procedure we recover the corresponding inequalities for the exponential distribution. These results have applications to interacting spin systems under a geometric reference measure.Mathematics Subject Classification. 60E07, 60E15, 60K35.

Binomial-Poisson entropic inequalities and the M/M/∞queue

“…, which gives (22). Finally, the correct constant for (P2) comes from the fact that 2A Φ = B Φ in that case.…”

“…The M/M/∞ queue has non-homogeneous independent increments, and is thus beyond this framework. The reader may find various entropic inequalities for finite space Markov processes in [17,33] and [2], and for infinite countable space Markov processes in [30], [1], [35], [29], [12], [22], [13,5], [11], [14,15,16], [21] for instance.…”

Learning to Rank under Multinomial Logit Choice

“…We begin with Theorem 2, whose proof is given in Appendix B which gives a deviation inequality for a function of multivariate Geometric data. The derivation of this result is based on theory from Bobkov and Ledoux (1998) and a logarithmic Sobolev inequality of Joulin and Privault (2004). Before stating our result we introduce a notion of the smoothness of a discrete function expressed in terms of its finite differences.…”

“…The first step of the proof is derive a bound on the relative entropy of functions with respect to the product measure. We make use of the following result from Joulin and Privault (2004). It gives a logarithmic Sobolev inequality tailored to functions of a univariate Geometric distribution.…”

“…Theorem 3 (Theorem 3.7 (Joulin and Privault, 2004)) Let π denote the law of a Geometric random variable with parameter p. Let 0 < b < − log(1 − p) and let f :…”