Choquet期望下极限定理及其收敛速度的刻画

The basis of the classical probability theorey is the assumption that probabilities or expectations are additive or linear. However, in many application fields, such as finance and robust statistics, the traditional additive probabilites/expectations fail to provide adequate information to describe or interpret the uncertain phenomena accurately. For this reason, more and more researchers turn to use nonlinear probabilities/expectations, a nature extension of the traditional ones, to model the uncertainty when the assumption of additivity is suspect. As a typical nonlinear expectation, Choquet expectation is famous for its applications in finance and ecnomics, therefor, the research on limit theorems under Choquet expectations becomes a heated subject worldwide. However, most of the exsiting results are based on the assumptions that the random variables are IID or independent with finite high moments conditions.. This project aims to establish the non-additive version of law of large numbers(LLN) under Choquet expectations with weaken assumptions. Further, we characterize its convergent accuracy under Choquet expectations with different moment conditions. Moreover, since we introduce the new notions of convolutionary independence and finite common moment conditions to replace the independence and identical distribution/high moment conditions of random variables separately, our LLN and its convergent accuracy can be naturally viewed as an extension of the traditional ones, also can be well appiled in financial fields due to the good properties of Choquet expetations.

数学期望具有线性性质是经典概率论的假设基础。然而，金融经济学、统计学等领域中存在着大量不确定现象，难以用线性概率或线性数学期望给出精准的刻画和预测。为此，人们提出了以具有非线性性质的数学期望为基础的非线性期望理论，以期能更准确地刻画带有不确定性的实际问题。作为一类典型的非线性数学期望，Choquet期望因其在金融等领域的广泛应用备受关注，因而对Choque期望理论的核心内容——极限定理的研究已逐步成为当今热点。但这些研究成果大多依赖于随机变量满足容度下事件独立性和同分布/高阶矩有限的假设条件。 . 本项目旨在通过引入Choquet期望下的卷积独立性和有限共一阶矩概念，以分别代替上述独立和同分布/高阶矩假设，从容度和期望两个角度依次建立更具一般性的Choquet期望下大数定律，探讨二者关系，并在不同阶矩条件下给出收敛误差估计，以清晰刻画其收敛精度。