Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications

Abstract: Classical Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers. In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng (2006, 2008b), we introduce the concept of negative dependence of random variables and establish Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations. As an application, we show that Kolmogorov's strong law of larger numbers holds for independent and identically distributed under a continuous sub-linear expectation if and only if the corresponding Choquet integral is finite.