I am studying the Stratonovich Integral and on wikipedia, Stratonovich Integral, it states that the integral, for a process $X:[0,T] \times\Omega \to \mathbb{R}$, as:
$$ \int_0^T X_t \circ dW_t $$
is defined as the limit in mean square of:
$$ \sum_{i = 0}^{k - 1} {X_{t_{i+1}} + X_{t_i}\over 2} \left( W_{t_{i+1}} - W_{t_i} \right) $$
What I don't understand is what is the meaning of "limit in mean square". From what I understand, we have that:
$$ lim_{k \to \infty}\sum_{i = 0}^{k - 1} {X_{t_{i+1}} + X_{t_i}\over 2} \left( W_{t_{i+1}} - W_{t_i} \right) = \int_0^T X_t \circ dW_t $$
Isn't this just the simple limit? Where is the mean square part come in, which by definition should have an expectation somewhere? I'd appreciate any insights, thanks!
It means that $$ lim_{k \to \infty}\Bbb E\left\{\left[\sum_{i = 0}^{k - 1} {X_{t_{i+1}} + X_{t_i}\over 2} \left( W_{t_{i+1}} - W_{t_i} \right) - \int_0^T X_t \circ dW_t\right]^2\right\}=0. $$