Moment Estimation Implied by the Bobkov-Ledoux Inequality

In this paper we consider a probability measure on the high dimensional Euclidean space satisfying Bobkov-Ledoux inequality. Bobkov and Ledoux have shown in (Probab Theory Related Fields 107(3):383–400, 1997) that such entropy inequality captures concentration phenomenon of product exponential measure and implies Poincaré inequality. For this reason any measure satisfying one of those inequalities shares the same concentration result as the exponential measure. In this paper using B-L inequality we derive some bounds for exponential Orlicz norms for any locally Lipschitz function. The result is close to the question posted by Adamczak and Wolff in (Probab Theory Related Fields 162:531–586, 2015) regarding moments estimate for locally Lipschitz functions, which is expected to result from B-L inequality.