Consider $B$ a Brownian motion, $0\leq T\leq\infty$, $H\in\Lambda^2_{loc}(T)$ (i.e. $H$ is progressive and $\int_0^T H_s^2ds <\infty$ a.s.), and the Ito integral $$ \int_0^T H_s dB_s. $$ Also, for a sequence of partitions $\pi^n=(t^n_0,\dots, t^n_{k_n})$ of $[0,T]$ with $mesh(\pi^n)\to 0$ consider the Riemann-Ito-Approximations $$ I^n = \sum_{i=0}^{k_n-1} H_{t^n_i}(B_{t^n_{i+1}} - B_{t^n_i}).\quad (\star) $$
I know that $I_n\to \int_0^T H_s dB_s$ in $L_2$ if $H$ is continuous and $\sup_{0\leq t\leq T}\vert H_t\vert$ is in $L^2$. What are some other conditions on $H$ (possibly together with conditions on $(\pi^n)$) for which the result still holds? If we weaken the mode of convergence (e.g. ask merely for the convergence to be in distribution), can the conditions be substantially generalized?
In particular, when is it true for $H$ deterministic and square-integrable, i.e. $H\in L^2([0,T])$? When does $\sum_{i=0}^{k_n-1} H_{t^n_i}\mathbb{1}_{[t^n_{i+1},t^n_i]}\to H$ in $L^2$ hold true?