Suppose $Z=(z_{1},...,z_{m})^T$ is m dimensional vector, each $z_{i}$ is independent identical distribution with mean 0. If we do linear transformation like: $$X = \Gamma Z$$ where $\Gamma$ is $p*m(p<m)$ matrix, $\Sigma = \Gamma \Gamma^T $.
Then will the independent $X_{1}$ sample from $X$, like $X_{1}^{T}X_{1}/\sqrt{p}$ be normal distribution when p goes to infinity? If not, what condition need for $\Sigma$?
Thanks so much for your help~