Asymptotic Behaviors For Dependent Bernoulli Variables

Probability limit theory is an important theoretical basis in probability and statistics,and it is of great significance to study the limit properties of random variable sequences.The first part is concerned with the background of the subject,the significance and the main research content of this paper.And then the models studied in this paper are also introduced.A typical Bernoulli process is a stochastic process of discrete sequence,which is independent and identically distributed and formed by some Bernoulli trials with two possible outcomes.In reality,weakening the independence of random variable sequence is more conducive to wide application.The generalized binomial distribution was proposed by Drezner and Farnum in 1993,and it was from a correlated Bernoulli process with interesting asymptotic properties which differed strikingly in the neighborhood of a critical point.The model of the generalized binomial distribution has been further generalized by some scholars,and some asymptotic results,such as central limit theorems,law of large numbers and law of the iterated logarithm and so on,have been established,which will be presented detailedly in the first chapter.In order to provide a description of the main content of this paper,the related basic knowledge will be described in detail.In the second chapter,we will introduce some relevant probability basics,such as the definition and property of Brownian motion and martingale,the law of large numbers,the central limit theorem,large deviations and so on.Then,some common lemmas of probability theory are introduced to prove the main results of the models.Our main results and proofs will be given in the third chapter.The aims of this chap-ter are to establish the strong laws of large numbers which extend some known results,and prove the moderate deviation principle for the different correlated Bernoulli models.In view of the complexity of dependent random variable sequence,we get our main results by a martingale representation.In this part,the limit theorems of partial sum are de-duced by the limit property of martingale.The convergence rate of the strong law of large numbers and the moderate deviation principle for the model are established by Hoeffding's inequality,Kronecker's lemma,Borel-Cantelli lemma,the property and the moderate de-viation principle of the martingale differences and so on.Furthermore,we generalize the parameters of the second model and obtain a series of asymptotic results in special cases.