Suppose I have $X$, a normal random variable with mean $\mu$ and variance $\sigma^2$. Now I discretise this random variable to form a discrete random variable $Y=g(X)$. $Y$ could be created by splitting the entire support set of $X$ into equal intervals, integrating $f_X(x)$ over each of those intervals and assigning that probability to the midpoint of the interval.
What nice properties of the Gaussian variable $X$ would $Y$ retain? For instance, if $Y_1$ and $Y_2$ are obtained by discretising two jointly Gaussian variables $X_1$ and $X_2$, will uncorrelatedness of $Y_1$ and $Y_2$ imply their independence?
I would appreciate if someone could lead me to some study/theory of such variables.