CLT for random quadratic forms based on sample means and sample covariance matrices

In this paper, we use the dimensional reduction technique to study the central limit theory (CLT) random quadratic forms based on sample means and sample covariance matrices. Specifically, we use a matrix denoted by $U_{p\times q}$, to map $q$-dimensional sample vectors to a $p$ dimensional subspace, where $q\ge p$ or $q\gg p$. Under the condition of $p\mathrm{/}n\to 0$ as $(p,n)\to \mathrm{\infty }$, we obtain the CLT of random quadratic forms for the sample means and sample covariance matrices.