| In this paper,we extend the CLT of sample spiked eigenvalues in generalized spiked covariance model proposed in Jiang and Bai(2021)[25]to the case RDS free,i.e.,there is no restriction to the existence of the limit of p/n,the Ratio of Dimension to sample Size(RDS),except an upper limit of the RDS to guarantee the spiked eigenvalues are distant.The choice of dimensionality and sample size is more flexible in our regime.Our new result not only removes the restriction on the existence of the limitation of RDS for the CLT of spiked eigenvalues of sample covariance matrix under usual high-dimensional settings,but also covers those under the so-called ultra-high dimensional regime as special cases.In order to better clarify the main theorem,this paper is divided into five chapters.Chapter 1 briefly introduces the background,the practical significance of the main theorem and some symbols.In Chapter 2,we introduce the development of spiked model and some famous and widely used conclusions in existing literature,including the asymptotic limit,the phenomenon of "phase transition" and the central limit theorem for spiked eigenvalues of covariance matrices.In addition,we define the generalized spiked covariance model,which is considered in this article.It’s a general non-negative definite matrix and allows the largest spiked eigenvalue tending to infinity.In chapter 3,RDS free central limit theorem for distant spiked eigenvalues of covariance matrices is proposed,besides displaying the assumptions,the detailed proofs and simulation experiments and the applications of the main theorem.In Chapter 4,the Generalized Fourth Moment Theorem(GFMT)and Partial Generalized Fourth Moment Theorem(PGFMT)are proposed.The Partial Generalized Fourth Moment Theorem is used to further prove the universality of the main central limit theorem,which is independent of distributions of samples.In Chapter 5,we make a summary of the contributions of this paper and give a brief description of the work we can do and will do in the future. |
