I'm working on this problem for my stochastic analysis course:
For $t \geq 0,$ let $W_t$ be a Wiener process defined on $( \Omega, \mathcal F, \mathbb{P})$, and let $$Q_{\Pi} = \sum_{i=0}^{n-1} (W_{t_{i+1}}-W_{t_{i}})^3.$$ Also, let $|| \Pi || = \text{max}_{0 \leq i \leq n-1}(t_{i+1}-t_i).$
Prove that as $|| \Pi || \rightarrow 0$, $\text{ } \mathbb{E}(Q_\Pi^2) \rightarrow 0.$
To be honest, I am not confident in my proof. Here is what I have so far; my strategy is to factor out a $(W_{t_{i+1}}-W_{t_{i}})^2$ and bound it by $||\Pi||$:
$$\mathbb{E}\big(Q_{\Pi}^2\big) = \mathbb{E}\bigg( \sum_{j=0}^{n-1} \sum_{i=0}^{n-1} (W_{t_{i+1}}-W_{t_{i}})^3(W_{t_{j+1}}-W_{t_{j}})^3 \bigg)$$
$$\leq \mathbb{E}\big((W_{t_{i+1}}-W_{t_{i}})^2\big) \mathbb{E}\bigg( \sum_{j=0}^{n-1} \sum_{i=0}^{n-1} (W_{t_{i+1}}-W_{t_{i}})(W_{t_{j+1}}-W_{t_{j}})^3 \bigg)$$
$$= (t_{i+1}-t_{i}) \cdot \mathbb{E}\bigg( \sum_{j=0}^{n-1} \sum_{i=0}^{n-1} (W_{t_{i+1}}-W_{t_{i}})(W_{t_{j+1}}-W_{t_{j}})^3 \bigg)$$
$$\leq ||\Pi|| \cdot \mathbb{E}\bigg( \sum_{j=0}^{n-1} \sum_{i=0}^{n-1} (W_{t_{i+1}}-W_{t_{i}})(W_{t_{j+1}}-W_{t_{j}})^3 \bigg),$$
which should approach zero as $||\Pi|| \rightarrow 0$.
How could I prove the implied statement in the problem? i.e., what am I missing? Is there a better way to approach this?
EDIT
Per Youem's suggestion in the comments, I've decided to Expectational value where i=j. This is what I get: $$(Q^2_Π)=\sum_{i=0}^{n−1}((W_{t_{i+1}}−W_{t_i})^6)$$
Not entirely sure where to go from here.
Let $\mu = \mathbb E [Z^6]$ for $Z\sim\mathcal N(0,1)$. Since $W_{t_{i+1}} - W_{t_i} \sim \mathcal N(0, t_{i+1} -t_i)\sim (t_{i+1} - t_i)^\frac12\mathcal N (0,1)$, $$\mathbb E \left[ \left( W_{t_{i+1}} - W_{t_i}\right)^6 \right] = \mu (t_{i+1}- t_i)^3 \le \mu \left\|\mathbb \Pi\right\|^2 (t_{i+1}-t_i)$$ then $$\sum_{i=0}^{n-1} E \left[ \left( W_{t_{i+1}} - W_{t_i}\right)^6 \right] \le \mu \left\|\mathbb \Pi\right\|^2 (t_{i+1}-t_i) = \mu \left\|\mathbb \Pi\right\|^2 \sum_{i=0}^{n-1}\left(t_{i+1} -t_i\right) = \mu\left\|\mathbb \Pi\right\|^2 (t_n - t_0)$$ then $$\mathbb E[Q_\Pi^2] \le \mu \Pi^2 (t_n -t_0) \le \mu \left\|\Pi\right\|^2 t$$