本文作为随机序列的收敛性、大数定律以及遍历定理等相关知识的读书报告, 从可测 函数序列的收敛概念谈起, 阐述了几乎处处收敛、依测度收敛、Lr 收敛、依分布收敛的定 义, 给出了一些判定是否收敛的结论. 另外讨论了四种收敛之间的强弱关系, 还给出了一些 反例, 用来说明某种收敛不能推出另外一种收敛. 在大数定律方面, 首先给出了 L2 收敛意义下的 Chebyshev 大数定律, 进而利用截尾以 及对称化的手法证明了弱大数定律成立的充分必要条件, 从而得到了 Khinchin 大数定理. 随后将弱大数定律推广到几乎必然收敛意义下的强大数定律. 最后采用列表的方式对比几 种大数定理的条件以及结论的差别, 并分析采用了什么样的手段使得大数定理的条件得以 减弱, 结论得以增强. 在遍历定理方面, 首先给出了平稳序列、保测变换的定义, 阐述了 Birkhoff 遍历定理. 利用极大遍历引理证明了 Birkhoff 遍历定理. 作为遍历定理的一个应用, 最后讨论了平稳序 列的常返性, 得到了 Poincare ´ 常返性定理.

This paper is a review of laws of large numbers, ergodic theorems and other related knowledge. Starting from the concept of convergence of measurable function sequence, the paper explained the almost sure convergence, converge in probability, Lr convergence and converge in distribution. This paper gave some criteria to determine whether a sequence of r.v. convergence. We also discussed the relationship between four kinds of convergence, some examples are given to illustrate that some kind of convergence unable to launch the other kind. On the laws of large numbers, our first set of weak laws called Chebyshev’s large number theorem convergence in L2 or in probability. Using the truncation and symmetry method, we obtained the Khinchin’s large number theorem. Then the weak law of large numbers is extended to the strong law of large numbers in which almost sure converge. Finally, we listed the comparison of several large number theorems’ condition and conclusion, and analyzed by what means large number theorem conditions can be reduced or the conclusion can be strengthened. On the ergodic theorem, we gave the definition of stationary sequence and measure preserving map firstly. Elaborated the Birkhoff’s ergodic theorem and proofed this result using maximal ergodic lemma. As an application of ergodic theorem, we finally discussed the recurrence of stationary sequence and obtained the Poincare’s recurrence theorem. ´