Let $(W_t)_{t\ge0}$ be a standard Brownian motion. Denote $\mathcal{P}:=\{0=t_1<t_2<\cdots<t_n=T\}$ and $\|\mathcal{P}\|:=\max_j|t_j - t_{j-1}|$. I would like to show that
$$
V^3(W_t):=\lim_{\|\mathcal{P}\|\to0}\sum_{j=1}^n(W_{t_j} - W_{t_{j-1}})^3\to0\quad{\text{in}}\quad L^2.
$$
I am able to prove this for convergence in probability (see https://quant.stackexchange.com/questions/17827, for example), but am struggling with the $L^2$ case.
EDIT: I've been a bit silly - it just occurred to me that the link I posted implicitly proves the $L^2$ convergence of the limit:
\begin{align*} \mathbb{E}\left[\left(\sum_{j=1}^n (W_{t_j}-W_{t_{j-1}})^3\right)^2\right] &= \sum_{i,j=1}^n \mathbb{E}\left[(W_{t_i}-W_{t_{i-1}})^3(W_{t_j}-W_{t_{j-1}})^3\right]\\ &=\sum_{j=1}^n \mathbb{E}\left[(W_{t_j}-W_{t_{j-1}})^6\right]\\ &=15\sum_{j=1}^n (t_j-t_{j-1})^3\\ &\leq 15T\|\mathcal{P}\|^2\to0\quad\text{as}\quad\|\mathcal{P}\|\to0. \end{align*}
However, I would also be interested to know if there is a simple proof for $V^p(W_t)$ with $p>2$, and whether the result also holds with probability one. Any hints or references would be greatly appreciated.
You can use that $|\sum_{i=1}^n(W_{t_{i+1}}-W_{t_i})^3|\leq \sup|W_{t_{i+1}}-W_{t_i}|\sum_{i=1}^n(W_{t_{i+1}}-W_{t_i})^2 \rightarrow 0$ since we have standart properties of the quadratic variation.
If I remember correctly the convergence is in $L^2$ but feel free to correct me