Limit theory: From linear expectation to nonlinear expectation

In the study of probability theory, the limit theory holds a pivotal position. Under the axiomatic system of Kolmogorov, from the initial Bernoulli's law of large numbers to the general form of the central limit 定理, the 假设 of independent and identically distributed (IID) for random variables is crucial in the development of probability theory. At this time, the central limit 定理 and the corresponding normal distribution occupy a “central" position in probability theory. However, when random variables do not satisfy the 条件 of independent and identical distribution, mathematicians have found that the asymptotic distribution is likely no longer a normal distribution. Therefore, exploring the limited distribution of the non-IID case has been an open question for more than two hundred years, and has attracted the attention of many scholars. In addition, from the perspective of the economic market, the Kolmogorov probability theory framework can well quantify the inherent laws of market operation, but it is very difficult to accurately depict the external impact of human behavior on the economic market by applying the Kolmogorov axiomatic system. For 例, the three famous paradoxes in the economic indicate that the probability theory under the Kolmogorov axiomatic system has obvious limitations in practical application, and many uncertain phenomena cannot be accurately modeled using linear probability and linear expectations. This has promoted the development of probability from a linear to a nonlinear framework. With the rise of nonlinear probability, establishing generalized conclusions on the asymptotic distribution of the sum of random variables under the framework of nonlinear expectations and non-IID cases has become a possibility. In this paper, we mainly introduce the development history of limit theory under the classical framework, as well as the main results of laws of large numbers, central limit theorems, and laws of the iterated logarithm under the framework of nonlinear expectations.