Complete Integration Convergence And The Laws Of The Logarithm Under Sub-linear Expectations

The probability limit theory has always played a very important role in the development of statistics.Driven by the demands of practical applications in the field of financial risk and insurance,the probability space can no longer meet the needs of market changes.The concept of sub-linear expectations is relatively completed,which solved the lack of classical probability space in the financial field.In this paper,we study the complete integration convergence for array of row-wise END random variables and the law of the logarithm of the ND random variables in a sub-linear space.First,this paper studies the complete integration convergence for array of row-wise END random variables under sub-linear expectations based on the complete moment convergence in probability space.Using methods such as variable substitution,censoring,piecewise summation and swapping sum order,with the help of different from the classical profile tools such as Rosenthal’s inequality under sub-linear expectations.This article shows that the results of complete moment convergence in probability space can be generalized to complete integral convergence under sub-linear expectations.Not only established the scope of the complete integral convergence results,but also widened the application field of the array of row-wise END random variables.Secondly,this paper used new exponential inequalities,moment inequality and capacity formulas under sub-linear expectations.It combined with the properties of sub-linear expectations,and used continuous local Lipschitz functions to modify the illustrative functions for processing,and combines inequality processing methods.Techniques,sub-column methods and other methods have studied the law of logarithm of the ND random variables under sub-linear expectations.In the proof process,the upper expectations inequality cannot prove that the law of the iterated logarithm obtained under the same conditions.Therefore,this paper extends law of the iterated logarithm of the NA random variables in the probability space to the law of the logarithm of the ND random variables under the sub-linear expectation,which makes the scope of research wider,generalizes and improves the corresponding results that enrich the limit theory of the law of the logarithm under sub-linear expectations.Since the non-additivity of the expectations and tolerances in the sub-linear expectations space,many research tools in the probability space are not applicable,resulting in more complicated and difficult research methods.This article is to overcome these difficulties to research the the complete integration convergence for array of row-wise END random variables and the laws of logarithm of the ND random variables under the sub-linear expectations.