Binary Uncertainty Bernoulli Model And Its Related Limit Theorems

In this thesis we will introduce a concise model to study the distribution uncer-tainties in the context of nonlinear probability,which we call the Binary Uncertainty Bernoulli Model.Furthermore,we prove a series of limit theorems about the model,including the law of large numbers,the large deviation principles and the central limit theorem.Especially,we study the central limit theorem from the perspectives of mean uncertainty and variance uncertainty,and give the explicit expression of its limit distri-bution,which provides a theoretical basis for the application of the binary uncertainty Bernoulli model in nonlinear probability and statistics.Finally,we establish the con-nection between the model and the "two-armed bandit problem" in statistical decision theory,which provides a new idea for studying the two-armed bandit problem from the perspective of nonlinear probability.In 1921,the American economist Frank Knight pointed out that in economics,probability and statistics models have unpredictable distribution uncertainty.Later this uncertainty is also called Knightian uncertainty.Knight believed that the "uncertainty"described by a single probability measure should be called risk,and the distribution uncertainty in the probability model should be described by a family of probability mea-sures more appropriately.To solve this problem,nonlinear probability and expectation theory came into being.There are currently two main research schools in this field:one is the capacity theory proposed by French mathematician Choquet[20]with "nonlin-ear measure" as the core;the other is the sub-linear expectation theory proposed by Professor Peng Shige[73],the academician of the Chinese Academy of Sciences,with"nonlinear expectation" as the core.But both the capacity theory and the sub-linea-expectation theory are establishing a more general axiomatic system.It seems that there is a lack of a basic model similar to the classic Bernoulli trials to help us understand nonlinear probability.We consider whether we can establish a "Bernoulli-like model in the context of nonlinear probability" to help us understand nonlinear probability theory more intuitively,so as to better study the distribution uncertainty.It is well known that,as the most basic probability model in traditional linear prob-ability theory,the Bernoulli trials have two basic properties:the trials are independent and identical distributed.Naturally,to construct a model with distribution uncertainty,the simplest case should be that there are only two possible distributions for each trial.Then the decision-maker will face a choice,which often relies on past experience,so there may no longer be independence between trials,and the results of previous trials may affect the distribution of subsequent trials.Based on the above characteristics,we construct a "Bernoulli-like model with distribution uncertainty" under the framework of nonlinear probability,and call it Binary Uncertainty Bernoulli Model.In order to make the model better applied in nonlinear probability and statistics,this thesis further stud-ies its basic properties and related limit theorems,including the law of large numbers,the large deviation principles,and the central limit theorem.It is worth noting that how to give an explicit expression for the limit distribution of the central limit theorem has always been a difficult and hot issue in the field of nonlinear probability research.This thesis uses Bang-Bang Brownian motion and oscillating Brownian motion to give explicit expressions for the limit distribution of the central limit theorem with mean uncertainty and variance uncertainty respectively.This is a result that is quite different from the existing nonlinear central limit theorem.In the process of researching this model,the "two-armed bandit problem" in statis-tical decision theory(abbreviated as TAB problem)(see[4,82])gave us great enlight-enment.The prototype of the TAB problem refers to a gambler operating a gambling game machine with two arms.When the gambler pulls one of the arms,he may get rewards or nothing.The probability distributions of the returns generated by the two arms are independent,and in general,the gambler does not know these two probability-distributions.The TAB problem is to design a strategy in this situation where informa-tion is limited or every choice is faced with uncertainty,so that the gambler can obtain the maximum expected return after n operations.In recent years,the TAB problems has many new applications and developments in biological modeling,data processing,machine learning and some other fields(see[42,52,88,91]).But as far as we know,the current research on this problem is based on the traditional linear probability theory.From the above description,we can see that the essence of the TAB problem is the prob-lem of distribution uncertainty.A natural idea is:will there be new breakthroughs in studying TAB problem under the framework of nonlinear probability?To this end,this thesis establishes the connection between the binary uncertainty Bernoulli model and the TAB problem,and discusses the asymptotic behavior of the TAB problem through the related limit theorems of the binary uncertainty Bernoulli model.Although this thesis does not give the optimal strategy to solve the TAB problem,we hope that it provides a new idea for studying the TAB problem.The dissertation is organized as follows:In Chapter 1,we establish the binary uncertainty Bernoulli model and study its basic properties.First,we explain the research background and construction ideas of the model.The binary uncertainty Bernoulli model describes a type of random exper-iment with distribution uncertainty.Each trial has two possible distributions.There is no longer independence between trials.The results of previous trials may affect the distribution of subsequent trials.Then,we use the language of nonlinear probability to give a strict mathematical definition of the model.We use a set of two-element proba-bility measures to describe the uncertainty of the distribution of each trial,and use the probability kernel to describe the dependence between the trials.Finally,we also obtain a series of important properties about the model,which lay the foundation for the study of the related limit theorems in the subsequent chapters.In Chapter 2,we mainly study the law of large numbers and the large deviation principles of the binary uncertainty Bernoulli model.First,we prove the weak law of large numbers of the model,and the result shows that the sample mean no longer converges to a fixed expected value,but falls between the maximum and minimum expectations of the random trials in the sense of the minimum probability.Then,we give the large deviation principle of the model.In particular,we give an explicit expression of its rate function.In Chapter 3,we study the central limit theorem of the binary uncertainty Bernoulli model from the perspective of mean uncertainty.Our results show that the limit distribution can be described by g-expectation or Bang-Bang Brownian motion under different test functions.Inspired by the results of Chen and Epstein[11],we still consider the combination of the law of large numbers and the central limit theorem,that is,the "statistic" is(?)(see(3.2.3)),to study the central limit theorem with mean uncertainty.The first part of this chapter proves that the maximum distribution of the "statistic" Tn,nQ still converges to g-expectation under this model.The second part proves that for a kind of symmetric function,the maximum distribution of the "statistic" Tn,nQ converges to Bang-Bang Brownian motion.Different from the result of Chen and Epstein[11],Bang-Bang Brownian motion has an explicit probability density function,which is convenient for application.In addition,their proof needs to use the theory of backward stochastic differential equations and partial differential equations,and our proof only needs to use the probability density function of Bang-Bang Brownian motion to perform simple calculus calculations,and use Lindeberg’s exchange ideas in traditional probability theory.Next,we removed the dependence of the statistic on the probability measure Q,and constructed a statistic that only depends on the sample data(?)(see(3.3.20)).Its maximum distribution still converges to Bang-Bang Brownian motion,which pro-vides a theoretical basis for the application of our model in nonlinear statistics.At the end of this chapter,as an application,we give an explicit expression of g-expectation and provide a method to simulate the probability distribution of Bang-Bang Brownian motion.In Chapter 4,we mainly study the central limit theorem with variance uncer-tainty for the binary uncertainty Bernoulli model,and the limit distribution can be described by the G-normal distribution and oscillating Brownian motion,under different test functions.The first part of this chapter proves that under our model,for the general test function,the maximum distribution of statistic(?)converges to G-Normal distribution.The second part proves that for a class of S-shaped functions(includ-ing unilateral indicator function,S-shaped utility function in prospect theory,etc.),the maximum distribution of statistic(?)converges to oscillating Brownian motion.Different from the G-normal distribution,the oscillating Brownian motion has an ex-plicit probability density function,which is easy to calculate and is conducive to the application of our model in the fields of economics,finance,probability and statistics.Finally,as an application,we give the explicit distribution function of the G-normal dis-tribution under a class of S-shaped utility functions and obtain a method for simulating the probability distribution of oscillating Brownian motion.In Chapter 5,we establish the connection between the binary uncertainty Bernoulli model and the TAB problem.We first prove that maximizing the expected utility of all strategies in the TAB problem is equivalent to maximizing the expected utility of all measures in the binary uncertainty Bernoulli model.Furthermore,using the limit theo-rems given in the previous chapters,we also discuss the asymptotic behavior in the TAB problem,which provides an idea for studying the TAB problem from the perspective of nonlinear probability.In Chapter 6,we summarize the main results and contributions of this thesis,and give some directions for future research.