Research On Limit Theory For Nonlinear And Linear Expectations

This thesis investigated some limit theory both in nonlinear expectations framework and linear expectations framework.Since Bernoulli proposed the famous "Bernoulli's law of large numbers" in 1713,limit theory has gradually developed into the central topic of probability theory.As one of the classic achievements of probability theory,limit theory ensures the rationality of statistical inference and has become the cornerstone of statistical inference,model prediction and data modeling.In many biostatistics and economic models,the nature of parameter estimation and hypothesis testing is exactly the combination of statistical methods and probability theory.The first section of this thesis explored the Cramer-type mederate deviations for the estimator of the drift coefficients of a class of non-stationary Ornstein-Uhlenbeck processes.The frequent outbreaks of international financial crises have caused a snowball effect on risk management problems of large financial institutions,and lead to high and urgent requirements for the risk assessment and management of financial derivatives.In 1921,Knight,the American economist,put forward the concept of "uncertainty" in his book"Risk,Uncertainty,and Profit",which was called "Knightian Uncertainty",sometimes also called "ambiguity".Knightian uncertainty told us that the probability faced by market behavior is not unique,so a single probability(linear probability)cannot be used to measure the risk(uncertainty in fact)in the market appropriately.Along with massive explorations and researches,the nonlinear elements were used to describe and explain the law of market widely.The second part of this thesis studied the limit theory about nonlinear expectations and obtained the corresponding convergence rate.This thesis was divided into six chapters.The main framework and results were as follows:In Chapter 1,we introduced and reviewed some basic concepts and important results about probability spaces,Choquet expectation spaces,Peng nonlinear expectation spaces and upper expectation spaces,including identical distribution,independence,law of large numbers,central limit theorem and deviations theory.In Chapter 2,we studied the Cramer-type moderate deviations for the estimator of the drift coefficient about non-stationary Ornstein-Uhlenbeck processes.Specifically,given a collection of independent standard Brownian motions {wk;k?1} on a complete probability space(?,F,{Ft}t?0,P),For any fixed N,the Ornstein-Uhlenbeck processes?uk;1?k?N} satisfy duk(t)=??k2?uk(t)dt+??k-?dwk(t),uk(0)=(U0,hk)=u0,t?0,1?k?N,where ?<0 was an unknown parameter,and ?>0,??0,??R\{0}were available.Moreover,?k>0 was known for any k.In this chapter,the Cramer-type moderate deviations for the maximum likelihood estimator of ?,in the above equation,was obtained by the property of multiple Wiener-It? integrals and asymptotic analysis techniques,in non-stationary cases.In Chapter 3,we proved the large deviations for random variables in upper expectation spaces taking values in Rd.Our conclusion has no requirement on the independence of random variables.This chapter introduced basic knowledge of large deviations in upper expectation spaces,in the first place,and then proved the large deviations of non-independent and identically distributed random variables in the upper expectation space.The last part of this chapter constructed a sequence of random variables in the framework of sublinear expectations satisfying our assumptions,to further elaborate and explain our results.In Chapter 4,we obtained the law of large numbers for weakly Fubini independent random variables in Ghoquet expectation spaces.First,the weak Fubini independent product measure and weak Fubini independent random variables were defined in the Choquet expectation space.Then,according to relevant properties obtained,the law of large numbers for weakly Fubini independent random variables was proved.It was important to note that,the Choquet expectation may not satisfy subadditivity,even for the subadditive capacity.In Chapter 5,we established two concentration inequalities,namely Bernsteintype inequality and McDiarmid's inequality,under upper probabilities.According to the conclusion of Bernstein's inequality,the convergence rates of the law of large numbers,the Marcinkiewicz-Zygmund type law of large numbers and the complete convergence,precise asymptotics about random variables in upper expectation spaces were obtained.In Chapter 6,we summarized the main content and innovation points of this thesis,and looked forward to the next step.