Some Convergence Properties Of Random Variable Sequences

The probability limit theory is one of the hot topics in modern probability theory.In this paper,some problems of probability limit theory are discussed,and we mainly study some convergence properties of random variable sequence.On the one hand,we further study almost sure central limit theorem in the traditional probability space.On the other hand,we study convergence properties of random variable sequences under the sublinear expectations,mainly studying strong law of large numbers,complete convergence,complete moment convergence of random variable sequences and self-normalized law of the iterated logarithm for geometrically weighted random series under the sub-linear expectations.Firstly,we study almost sure central limit theorem for products of sums of partial sums by using variable substitution and polynomial expansion of logarithmic function,estimating the covariance of variables,skillfully using truncation,subsection summation,the exchange of summation order and subsequence,etc.It is proved that the almost sure central limit theorem for products of sums of partial sums of independent random variable sequences is still valid for some kind of unbounded measurable functions.A general result of almost sure central limit theorem is obtained,which extends the scope of almost sure central limit theorem.Secondly,we study and obtain almost sure central limit theorem for self-normalized weighted sum of mixing sequences by using the self-normalized limit theory,the probability inequality of mixing sequences,central limit theorem and Slutsky theorem.In this theorem,we get a stronger conclusion by larger weight.So we obtain better results.Thirdly,we investigate strong convergence properties for weighted sums of random variable sequences(arrays)under the sub-linear expectations by using new moment inequalities,capacity formulas and exponential inequalities under the sublinear expectations,and fully combining the properties of sublinear expectations,skillfully using local Lipschitz function,using inequality processing techniques and subsequence methods,and so on.We establish complete convergence theorems and a complete moment convergence theorem for arrays of rowwise negatively dependent random variables under the sub-linear expectations.General strong law and complete convergence theorems are also obtained for weighted sums of extended negatively dependent random variables under the sub-linear expectations.Our results of strong limit theorems extend and improve the corresponding results in the traditional probability space.These results enrich the limit theory of sublinear expectation space.Finally,we investigate and establish a self-normalized law of the iterated logarithm for geometrically weighted random series of independent random variables under some minimal conditions under the sub-linear expectations by using Berstain inequality,skillfully truncating,transforming the limit of the weight,dealing with the tail of series,and skillfully using local Lipschitz function,and so on.Our result enriches the selfnormalized limit theory of the sub-linear expectation space.