(Quadratic variation) Let \((B(t))_{t \geqslant 0}\) be a one-dimensional Brownian motion. Consider the random variables \(Y_{n}:=\sum_{k=1}^{n}\left(B\left(\frac{k}{n}ight)-B\left(\frac{k-1}{n}ight)ight)^{2}\).

a) Find \(\mathbb{E} Y_{n}\) and \(\mathbb{V} Y_{n}\).

b) Determine the probability density of \(Y_{n}\).

c) Find the characteristic function \(\phi_{n}(\xi)=\mathbb{E} e^{i \xi Y_{n}}, \xi \in \mathbb{R}\), and determine the \(\operatorname{limit}^{2} \lim _{n ightarrow \infty} \phi_{n}(\xi)\).

d) Show that \(\lim \mathbb{E}\left[\left(Y_{n}-cight)^{2}ight]=0\) and determine the constant \(c\).