The Pseudo-Independence,Independence And Limit Theorems Under Sublinear Expectations

In order to deal with a series of financial problems concerning Knightian uncertainty,Academician Peng Shige pioneered the nonlinear expectation theory in 2006.As a generalization of the concept of independence in classical probability theory,the notion of independence under sublinear expectation was proposed by Academician Peng Shige,and based on it,the law of large numbers and the central limit theorem under sublinear expectations are established.In the past decades,the research about basic theory on nonlinear expectation has become more and more in-depth,and nonlinear expectation limit theory has obtained more and more abundant results.Moreover,nonlinear expectations have been increasingly widely used in the practical research in the field of finance.Based on the reconstruction of independence under nonlinear expectations,this thesis further improves the research about nonlinear expectation limit theorems.The main innovation lies in three points:Firstly,the concept of quasiindependence is proposed,and conditional expectation is introduced to examine the independence in the sublinear expectation space,exploring the limit distribution performance of random variable sequences in the sublinear expectation space;Secondly,a new method of constructing independent sequences is proposed,which provides a new idea for studying the limit distribution properties of independent sequences via the good properties of canonical spaces;The third point is to establish a central limit theorem under mean-uncertainty,and provide a new proof method-the martingale method-that is not based on the regularity estimation of partial differential equations.At the same time,the law of large numbers,the law of iterated logarithms and one-sided central limit theorem under quasi-independence are established,which have both theoretical and practical value.This thesis consists of six chapters.Chapter 1 Introduction briefly introduces the financial background and theoretical and practical significance of nonlinear expectation theory,the development of limit theorems mainly including the law of large numbers,central limit theorem,and the law of iterated logarithms,as well as the current research status of nonlinear expectation limit theory.Chapter 2 reconstructs independence in sublinear expectation spaces.Explain the concept of quasi-independence and explore the close relationship between quasi-independence and independence.Consider whether independence under nonlinear expectations implies independence in classical probability theory.A new construction method of independent sequences is given,and the connection between independent sequences in general sublinear space and independent sequences in canonical space is constructed.The third chapter mainly studies the law of large numbers under nonlinear expectations.The law of large numbers for quasi-independent non-identically distributed sequences is established under relatively loose moment conditions.A strong law of large numbers with bounded random variables as the limit is obtained in canonical space,and an attempt is made to apply the law of large numbers to nonlinear expections parameter estimation.In Chapter 4,a functional central limit theorem with mean uncertainty is established.In this chapter,we first prove a functional central limit theorem with mean uncertainty in canonical spaces based on the martingale characterization of G-Brownian motion,which is different from the commonly used methods relying on the regularity estimation of partial differential equations.Then,as an application of the construction of independent sequences in Chapter 2,we gradually extend it to regular sublinear expectation spaces and general sublinear expectation spaces.And as a corollary,the central limit theorem under the condition of quasi-independence is obtained.Chapter 5 focuses on the law of iterated logarithms under nonlinear expectations.In the non-degenerate case,we establish the law of iterated logarithm for quasi-independent non-identically distributed sequences,as well as the similar law under higher order moment conditions.Then,the law of the iterated logarithm containing the degenerate case is obtained in the canonical space and extended to the regular sublinear expectation space.The law of iterated logarithm with limits accessible everywhere is also yielded.The final chapter makes a summary of the full text and proposes ideas for future work.