Invariance principle of random projection for the norm

Johnson-Lindenstrauss guarantees certain topological structure is preserved under random projections when embedding high dimensional deterministic vectors to low dimensional vectors. In this work, we try to understand how random projections affect norms of random vectors. In particular we prove the distribution of norm of random vectors $X \in \mathbb{R}^n$, whose entries are i.i.d. random variables, is preserved by random projection $S:\mathbb{R}^n \to \mathbb{R}^m$. More precisely, \[ \frac{X^TS^TSX - mn}{\sqrt{\sigma^2 m^2n+2mn^2}} \xrightarrow[\quad m/n\to 0 \quad ]{ m,n\to \infty } \mathcal{N}(0,1) \]