$E \left\{ \left( \sum_{i=0}^{n-1} \left[ B_{c_i} \left( B_{t_{i+1}} - B_{t_i}\right)\right] \right)^2 \right\}$, where $c_i \in [t_i, t_{i+1}]$ Let $B$ be a standard Brownian motion and $\{t_i\}_{i=0}^n$ a partition of $[0,t]$.
Define $c_i= (1-c)t_{i+1}+ct_i$, for some $c \in [0,1]$.
Write $B_i$ for $B_{t_i}$ and
$$
S_n=\sum_{i=0}^{n-1}  B_{c_i} \left( B_{i+1}-B_i \right).
$$
Calculate
$$
\lim_{n \to \infty} E \left( S_n^2 \right).
$$
The only way I know is to use orthogonality of increment of Brownian motion.
That is segmenting each variable in the summand in term of increment, should deliver a sum $S_n^2$ which telescopes to a simple result.
I tried this calculation over and over, but my resulting expression doesn't telescope.
Is there another way to do this?
 A: It suffices to evaluate
$$ \Bbb{E} \big[ B_{c_i} \big( B_{t_{i+1}} - B_{t_{i}} \big) B_{c_j} \big( B_{t_{j+1}} - B_{t_{j}} \big) \big]. \tag{1} $$
To this end, we make the following simple observation:

Observation. For $w \leq v \leq u$,
$$\begin{align*}
\Bbb{E} \big[ B_{v} \big( B_{u} - B_{w} \big) \big]
&= \Bbb{E} \big[ B_{v} \Bbb{E} \big[ B_{u} - B_{w} \big\vert \mathcal{F}_{v} \big] \big]
= \Bbb{E} \big[ B_{v} \big( B_{v} - B_{w} \big) \big] \\
&= \Bbb{E} \big[ \big( B_{v} - B_{w} \big)^2 + B_{w} \big( B_{v} - B_{w} \big) \big] \\
&= \Bbb{E} \big[ \big( B_{v} - B_{w} \big)^2 \big] + \Bbb{E} \big[ B_{w} \Bbb{E} \big[ B_{v} - B_{w} \big\vert \mathcal{F}_{w} \big] \big]  \\
&= \vphantom{\big[} v - w.
\end{align*}$$

Now assume first that $i \neq j$. We may assume $i < j$. Then $t_{i+1} \leq t_j$ and
$$ \begin{align*}
&\Bbb{E} \big[ B_{c_i} \big( B_{t_{i+1}} - B_{t_{i}} \big) B_{c_j} \big( B_{t_{j+1}} - B_{t_{j}} \big) \big] \\
&= \Bbb{E} \big[ B_{c_i} \big( B_{t_{i+1}} - B_{t_{i}} \big) \Bbb{E} \big[ B_{c_j} \big( B_{t_{j+1}} - B_{t_{j}} \big) \big\vert \mathcal{F}_{t_j} \big] \big] \\
&= \Bbb{E} \big[ B_{c_i} \big( B_{t_{i+1}} - B_{t_{i}} \big) \Bbb{E} \big[ \big(B_{c_j} - B_{t_j} \big) \big( B_{t_{j+1}} - B_{t_{j}} \big) + B_{t_j} \big( B_{t_{j+1}} - B_{t_{j}} \big) \big\vert \mathcal{F}_{t_j} \big] \big] \\
&= \Bbb{E} \big[ B_{c_i} \big( B_{t_{i+1}} - B_{t_{i}} \big) \Bbb{E} \big[ \big(B_{c_j} - B_{t_j} \big) \big( B_{t_{j+1}} - B_{t_{j}} \big) \big\vert \mathcal{F}_{t_j} \big] \big] \\
&= \Bbb{E} \big[ B_{c_i} \big( B_{t_{i+1}} - B_{t_{i}} \big) \big] \Bbb{E} \big[ \big(B_{c_j} - B_{t_j} \big) \big( B_{t_{j+1}} - B_{t_{j}} \big) \big] \\
&= \big(c_i - t_i\big)\big(c_j - t_j\big) \\
&= (1-c)^2 \big(t_{i+1} - t_i\big)\big(t_{j+1} - t_j\big)
\end{align*}$$
By interchanging the role of $i$ and $j$, we find that this result holds for all $i \neq j$. Thus it remains to evaluate $(1)$ for the case $i = j$. But
$$\begin{align*}
&\Bbb{E} \big[ B_{c_i}^2 \big( B_{t_{i+1}} - B_{t_{i}} \big)^2 \big] \\
&= \Bbb{E} \big[ B_{c_i}^2 \big\{ \big(B_{t_{i+1}} - B_{c_{i}} \big)^2 + 2\big(B_{t_{i+1}} - B_{c_{i}} \big)\big(B_{c_{i}} - B_{t_{i}} \big) + \big(B_{c_{i}} - B_{t_{i}} \big)^2 \big\} \big] \\
&= \Bbb{E} \big[ B_{c_i}^2 \big] \Bbb{E} \big[ \big(B_{t_{i+1}} - B_{c_{i}} \big)^2 \big] + 2 \Bbb{E} \big[ B_{c_i}^2 \big(B_{c_{i}} - B_{t_{i}} \big) \Bbb{E} \big[ \big(B_{t_{i+1}} - B_{c_{i}} \big) \big\vert \mathcal{F}_{c_i} \big] \big] \\
&\qquad + \Bbb{E} \big[ B_{c_i}^2 \big(B_{c_{i}} - B_{t_{i}} \big)^2 \big] \\
&= c_i \big( t_{i+1} - c_i \big) + 0 + \Bbb{E} \big[ \big\{ \big( B_{c_i} - B_{t_i} \big)^2 + 2B_{t_i} \big( B_{c_i} - B_{t_i} \big) + B_{t_i}^2 \big\} \big(B_{c_{i}} - B_{t_{i}} \big)^2 \big] \\
&= c_i \big( t_{i+1} - c_i \big) + \Bbb{E}\big[ \big( B_{c_i} - B_{t_i} \big)^4 \big] + \Bbb{E} \big[ 2B_{t_i} \big] \Bbb{E} \big[ \big(B_{c_{i}} - B_{t_{i}} \big)^3 \big] + \Bbb{E} \big[ B_{t_i}^2 \big] \Bbb{E} \big[ \big(B_{c_{i}} - B_{t_{i}} \big)^2 \big] \\
&= c_i \big( t_{i+1} - c_i \big) + 3\big(c_i - t_i\big)^2 + t_i \big( c_i - t_i \big) \\
&= c_i \big( t_{i+1} - t_i \big) + 2(1 - c)^2\big(t_{i+1} - t_i\big)^2.
\end{align*}$$
Combining these results together, we have
$$ \begin{align*}
\Bbb{E} \big[ S_n^2 \big]
&= \sum_{i,j=0}^{n-1} \Bbb{E} \big[ B_{c_i} \big( B_{t_{i+1}} - B_{t_{i}} \big) B_{c_j} \big( B_{t_{j+1}} - B_{t_{j}} \big) \big] \\
&= \sum_{i,j=0}^{n-1} (1-c)^2 \big(t_{i+1} - t_i\big)\big(t_{j+1} - t_j\big) + \sum_{i=0}^{n-1} \big\{ c_i \big( t_{i+1} - t_i \big) + (1 - c)^2\big(t_{i+1} - t_i\big)^2 \big\} \\
&= (1-c)^2 t^2 + \sum_{i=0}^{n-1} c_i \big( t_{i+1} - t_i \big) + O\Big( t \max_{0 \leq i \leq n-1} \big( t_{i+1} - t_{i} \big) \Big) \\
&\xrightarrow[n\to\infty]{} (1-c)^2 t^2 + \int_{0}^{t} s \, ds = \left( \frac{1}{2} + (1 - c)^2 \right) t^2.
\end{align*}$$
