Assume that we have SDE:
\begin{align} dX_t = a(X_t)dt+b(X_t)dBt \end{align}
where $B_t$ is a standard Brownian motion.
For numerical simulation, we could try
\begin{align} X_{k+1}^{\Delta t} = a(X_k^{\Delta t})dt+\sqrt{\Delta t}b(X_k^{\Delta t})dB_t \end{align}
However, my question is whether that simulation will work even if we have a different distribution instead of $B_t$ ~ $N(0,1)$
I tested with (case 1: standard normal distribution) & (case 2: standard uniform distribution), but the result seems to be similar...
Maybe I made a mistake in my code, but does any independent standard non-normal distribution simulation still work theoretically??