The Marcinkiewicz-Zygmund Type Strong Law Of Large Numbers With General Normalizing Sequences Under Sublinear Expectations

As one of the basic limit theorems in classical probability theory,the law of large numbers plays an important role in the development and real applications of probability theory.In real life,many uncertain phenomenas cannot be modeled by additive probability and linear expectation,the application of the law of large numbers is limited by additive condition.Therefore it is very significant to study the generalization of the law of large numbers under the nonlinear framework,which has important theoretical and application significance.One of the widely studied nonlinear expectations is the sublinear expectations introduced by Peng.In recent years,under the framework of the sublinear expectations theory,more and more conclusions of the law of large numbers are studied.Because of the nonadditivity of sublinear expectations and capacities,common methods in classical probability space are no longer vaild,and the study of the law of large numbers under sublinear expectations becomes more complex and challenging.Inspired by the work of Anh et al.[2],we study the Marcinkiewicz-Zygmund type strong law of large numbers with general normalizing sequences under sublinear expectations.Based on Peng’s sublinear expectation,the notion of complete convergence of weighted sums of random variables,we prove the MarcinkiewiczZygmund type strong law of large numbers under the condition of finite seriesΣn≥Aαn-1L2ε(n1/α)<∞ and the moment conditionMost of current researches on the relevant conclusions under sublinear expectations are based on a special kind of slowly varying functions,and obtain the corresponding complete convergence conclusions.To the best of our knowledge,there is no research on complete convergence and the Marcinkiewicz-Zygrnund type strong law of large numbers based on a general slowly varying functions yet.In this paper,we study the M-Z type strong law of large numbers for general slowly varying functions,and the conclusion is valid for all slowly varying functions.