I have been working to prove the rank of a sample correlation matrix. Let $% X_{t}$ be $n\times k$ matrix of observation, where $n>k.$ We assume $% rank\left( X_{t}\right) =k,$ i.e., $X_{t}$ is column full rank for all $% t=1,...m$. Let $S_{XX}=\frac{1}{nm}\sum_{t=1}^{m}X_{t}X_{t}^{T},$ i.e., $% S_{XX}$ is a $n\times n$ matrix. Assuming both $n$ and $m$ are large and $k$ is finite, say, $n,m=100,$ and $k=5.\,$What's the rank of $S_{XX}?$ I tried some simulations and found the rank of $S_{XX}$ is $k,$ but I don't know how to prove it. Thanks.
There is a similar question What is the rank of correlation matrix and its estimate? claim the rank of $S_{XX}$ is $k$, but can we prove it? Thanks a lot!
