Teachers:
I have come across the following problem:
Suppose we have time series of $B_{i\Delta}$ with $i=0,1\ldots,n-1$ and $B_{t}$ is a standard Brownian motion.
How to prove the following results: (as $\Delta\to 0$, $n\to\infty$ and $n\Delta\to\infty$) \begin{eqnarray} \frac{B_{\left(n-1\right)\Delta}\sum\limits_{l=0}^{n-1} B_{l\Delta}}{\sum\limits_{l=0}^{n-1}\left(B_{l\Delta}\right)^{2}} &\overset{p}{\rightarrow}&0\,,\\ \frac{\left(B_{\left(n-1\right)\Delta}\right)^{2}}{2\Delta \sum\limits_{l=0}^{n-1}\left(B_{l\Delta}\right)^{2}}&\overset{L}{\rightarrow}&\frac{\int_{0}^{1}BdB}{\int_{0}^{1}B^{2}dr}\,, \end{eqnarray} where $\overset{p}{\rightarrow}$ means convergence in probability and $\overset{L}{\rightarrow}$ means convergence in distribution.
Appricating for your kindly help.