The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

Let {X_n;n≥1} be a sequence of independent random variables on a probability space（Ω,F,P） and S_n=∑k=1~n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of S_n are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.