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Johnson-Lindenstrauss lemma states random projections can be used as atopology preserving embedding technique for fixed vectors. In this paper, wetry to understand how random projections affect probabilistic properties ofrandom vectors. In particular we prove the distribution of inner product of twoindependent random vectors $X, Z \in {R}^n$ is preserved by random projection$S:{R}^n \to {R}^m$. More precisely,  \[  \sup_t \left| \text{P}(\frac{1}{C_{m,n}} X^TS^TSZ <t) -\text{P}(\frac{1}{\sqrt{n}} X^TZ<t) \right| \le O\left(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{m}} \right)  \]  This is achieved by proving a general central limit theorem (product-CLT) for$\sum_{k=1}^{n} X_k Y_k$, where $\{X_k\}$ is a martingale difference sequence,and $\{Y_k\}$ has dependency within the sequence. We also obtain the rate ofconvergence in the spirit of Berry-Esseen theorem.

A product-CLT and its application in invariance principle of random projection