一些概率不等式及其在极限理论与统计中的应用研究

As in the other branch of mathematical science, in probability and statistics, inequalities play an important role and are regarded to be even more important than equalities. In the project, some inequalities will be extended, meanwhile, some applications will be given by using these and those inequalites, some new results will be obtained in probability limit theory and statiatics.The specific researches are: 1. Bahr-Esseen's type moment inequality will be extended to the dependent random variable case, and some applications are given in the mean convergence, the rate of the mean convergence, the rate of the inverse moment, the consistency of estimator by using Bahr-Esseen's type moment inequality; 2. by using the finite dimensional approximation of random element, the mean convegence will be discussed; 3. by using Ful-Nagaev's type inequality, under the condition that the sequence of random elements is bounded in probability, the similar strong law of large number, the complete convergence and the complete moment convergence related to the maximum partial sums will be discussed, the strong law of large number and the single law of logarithm for the delayed sums will be discussed, the complete convergence and complete moment convergence for the maximum partial sums relating to the law of the iteratied logarithm will be discussed, etc.

如同数学科学的其它分支一样, 在概率统计中, 不等式常常起着重要的甚至是关键的作用. 本项目一方面将推广一类不等式，另一方面将给出这类不等式和一些已有不等式的应用, 获得概率极限理论和统计中的一些新结果. 其具体研究内容是: 1.对相依随机变量序列的部分和建立Bahr-Esseen型矩不等式并在均值收敛性, 均值收敛速度，逆矩的收敛速度,统计量的相合性等方面给出应用; 2. 应用随机元的有限维逼近不等式, 研究随机元序列的均值收敛性等等; 3. 应用随机元的Fuk-Nagaev 型不等式, 在随机有界条件下, 研究随机元序列相对于强大数定律的强收敛性、部分和最大值的完全收敛性和矩完全收敛性等, 研究后置和的强大数定律和单对数律等, 研究相对于重对数律和中偏差的部分和的最大值的更精确的完全收敛性和矩完全收敛性, 以及相应的精确渐近性等.

如同数学科学的其它分支一样, 在概率统计中, 不等式常常起着重要的甚至是关键的作用. 本项目一方面将推广一类不等式，另一方面将给出这类不等式和一些已有不等式的应用, 获得概率极限理论和统计中的一些新结果. 其具体研究内容是: 1.对相依随机变量序列的部分和建立Bahr-Esseen 型矩不等式并在均值收敛性, 均值收敛速度，统计量的相合性等方面给出应用; 2. 应用随机元的有限维逼近不等式, 研究随机元序列的均值收敛性等等; 3. 研究随机元的Fuk-Nagaev 型不等式的应用, 研究NOD序列的Fuk-Nagaev 型不等式及应用.