Let $W_t$ be Wiener process. I am trying to evaluate the following limit $\lim\limits_{n \to \infty}~{\sum\limits_{i=1}^{n}W_{\frac{i-1}{n}+\frac{1}{2n}}\left( W_{\frac{i}{n}} - W_{\frac{i-1}{n}} \right)}$
I've expanded parethesis and got $ \lim\limits_{n \to \infty}~ \left[ -W_0W_\frac{1}{2n} - W_\frac{1}{n}\left( W_\frac{3}{2n} - W_\frac{1}{2n} \right) - \cdots - W_\frac{n-1}{n}\left( W_\frac{2n-1}{2n} - W_\frac{2n-3}{2n} \right) + W_\frac{2n-1}{2n}W_1 \right] $
I think I should apply central limit theorem here, but I don't understand how.