Some Results On Limit Theory For The Largest Entries Of Sample Correlation Matrices Under Mixing Assumptions

Classical limiting theorems in statistics are under the assumption that the dimension of sample is fixed,which have been found to be seriously inadequate in aiding in the analysis of very high dimensional sample.The theory of random matrices might be one possible method for dealing with large dimensional sample analysis and hence has received more attention among statisticians in recent years.The theory of random matrices has widespread applications in many research areas,such as signal processing,network security,image processing,genetic statistics,stock market analysis,and other finance or economic problems.The correlation matrix is an important statistic in the multivariate analysis,it plays pivotal roles in the statistical analysis of a multivariate sample,its maximum likelihood estimator is the sample correlation matrix.Therefore,it is of great significance to study the sample correlation matrix.Although there has been a large amount of literatures on the sample correlation matrix,most of the research is based on independent assumption.For a sample,it is reasonable to assume its independence,however sometimes it is quite difficult to verify the independence of a sample.Hence,it is necessary to study the limit theory of sample correlation matrix under the mixing assumptions.Based on the above discussion,this thesis presents the logarithmic law and the asymptotic distributions for the largest off-diagonal entries of sample correlation matrix under the mixing assumptions by using the methods of truncation of random variables,strong approximation and Chen-Stein Poisson approximation.In Chapter 2,we get a logarithmic law of the largest entries of sample correlation matrices under a φ-mixing assumption.We consider a p-variate population X,X_n×p is an n by p matrix,where the n rows are observations from a certain multivariate distribution and each of p columns is an n observation from a variable of the population distribution.The n rows of X_n×p are strictly stationary φ-mixing random vectors and each of p columns is an independent and identically distributed random vector.The logarithm law of L_n=max_1≤i<j≤p|（?）_ij| under a φ-mixing assumption is obtained by the method of the probability inequalities of φ-mixing random variables sequences,truncation of random variables,strong approximation and Chen-Stein Poisson approximation,where （?）_ij is the Pearson correlation coefficient generated by the ith and the jth column of X_n×p.In Chapter 3,we get a logarithmic law of the largest entries of sample correlation matrices under an α-mixing assumption.In this chapter,we consider an n by p matrix X_n×p where the n rows are strictly stationary α-mixing random vectors and each of p columns is an independent and identically distributed random vector.First of all,we mainly apply some probability inequality of the sequences of α-mixing random variables,inequality expansion and contraction are used to approximate α-mixing random variables by using independent random variables.Secondly,we approximate independent random variables by using normally distributed random variables,Then,the law of the logarithm for the largest off-diagonal entries of sample correlation matrices under an α-mixing assumption by Chen-Stein Poisson approximation method.In Chapter 4,we mainly study the asymptotic distributions of the largest entries of sample correlation matrices under an α-mixing assumption.In this chapter,we consider an n by p matrix X_n×p,where the n rows are strictly stationary α-mixing random vectors and each of p columns is an independent and identically distributed random vector.The methods of truncation of random variables and strong approximation,approximating sample correlation matrices by sample covariance matrices,Chen-Stein Poisson approximation and so on are used.According to the probability inequality of α-mixing random variables sequences,the asymptotic distributions of the largest entries of sample correlation matrices under an α-mixing assumption are obtained.On this basis,we prove the asymptotic distributions of the largest off-diagonal entries of sample correlation matrices under an α-mixing assumption is the extreme distribution of type I,which enriches the theoretical results of sample correlation matrix.