Convergence Properties For Asymptotically Almost Negatively Associated Random Variables Under The Sub-linear Expectations

In classical probability,the additivity of probability and expected value is assumed.But in fact,this additivity assumption is not feasible in many application fields,because the uncertainty phenomenon cannot use additivity probability or additivity expectation Modeling.Non-additive probability and non-additive expectation are useful tools for studying uncertainty in statistics,risk measurement,financial hedging,and nonlinear stochastic calculus.In recent years,the theory and methods of nonlinear expectations have been well developed.Good development,and has received extensive attention in application fields such as financial risk measurement and control.Professor Peng Shige introduced a typical example of nonlinear expectations in the framework of backward stochastic differential equations,called g-expectations.From above Since the 1990 s,the g-expectation based on the backward differential equation and its related properties have been extensively developed,and many practical problems in various fields have been solved.In addition,the limit theory currently studied is mainly based on the sequence and distribution of random variables The convergence of function sequences is an important research direction in probability theory and mathematical statistics.In practice,the occurrence of many random events is not independent,and the concept of dependence follows,and it is used in actuarial insurance,survival analysis and economics Decision-making and many other fields have been widely used.In this paper,first,we get the Hájek-Rényi type maximum inequality under sub-linear expectations,and obtain the strong law of large numbers for the partial sum of random variables under sub-linear expectations,and expanded this to obtain the strong law of large numbers and of the partial sum of asymptotically almost negatively associated random variable.Secondly,we investigate the complete convergence and complete moment convergence for maximal partial sums of asymptotically almost negatively associated random variables under the sub-linear expectations.The results obtained in the article are the extensions of the complete convergence and complete moment convergence under classical linear expectation space.