  Johnson-Lindenstrauss lemma states random projections can be used as a
topology preserving embedding technique for fixed vectors. In this paper, we
try to understand how random projections affect probabilistic properties of
random vectors. In particular we prove the distribution of inner product of two
independent 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 of
convergence in the spirit of Berry-Esseen theorem.
