| The linearity of probability and expectation in classical probability space makes them suitable for analysis and modelling of data with deterministic distribution.However,in the fields of economics and finance,there are many data with uncertain distribution,so the theory of probability space is no longer applicable to the analysis and simulation of these data with the distribution uncertainties.In order to solve these uncertainties,Academician Peng Shige proposed sub-linear expectation.This paper investigates limit theory based on the theory of sub-linear expectations.Theorems of complete integral convergence and almost sure convergence for dependent random variables under sub-linear expectations are established.In this paper,we study the sequence of END random variables and obtain that the weights are the complete convergence and complete integral convergence of randomly weighted sums of random variables;a wider sequence of WND random variables than END is also studied,the dominating coe cient is the regularly varying function,and the complete convergence and complete integral convergence of the weighted sums are obtained;and the arrays of WND random variables with arbitrary control coe cient is considered,and the complete convergence and complete integral convergence of its weighted sums are established.The obtained results compared with the existing results,the weight is extended from real sequence to random variable sequence,the range of random variables is expanded from END to WND,and the partial sum is promoted to weighted sum.In this article,almost sure convergence of END arrays under sub-linear expectation space is also studied.Since almost sure convergence in sub-linear expectation space is essentially different from the almost sure convergence in probability space: almost sure lower capacity convergence can be derived from almost sure upper capacity convergence,and the converse is not true,which makes it more di cult to study almost sure convergence under the sub-linear expectations.In this paper,an upper bound on the upper limit almost sure and a lower bound on the lower limit almost sure are established.When the upper expectation is equal to the lower expectation,we obtain an upper bound on the upper limit almost sure.The results extend the corresponding contents of the probability space,and the conditions are weaker and the conclusions more general than the corresponding results for the sub-linear expectation space.Since the essence of sub-linear expectation space is upper expectation,independence,identically distributed,and so on are defined in terms of upper expectation,and the equality of upper expectation does not mean equality of upper capacity.Furthermore,because the upper expectation and upper capacity are only sub-additive,the resulting capacity inequalities of partial sum are unilateral,the limit theory of sub-linear expectation space is fundamentally different from that of probability space.Thus,we study the limit theory from the unilateral side and then seek for the conditions for the transition to bilateral. |
