Sampled or discretised Gaussian Random Variables

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.

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Rounding to the midpoint makes the variance bigger, because the rounding error is actually negatively correlated with the random variable being rounded. This is the opposite of what happens with the uniform distribution, where the variance would become smaller, and the rounding error would be independent of the rounded random variable. – Michael Hardy Mar 15 '12 at 12:47
@Bravo: I can't migrate this question to MO and I'm also not convinced it's a good fit there. Perhaps wait a little longer for responses first. – Qiaochu Yuan Mar 16 '12 at 18:10
@QiaochuYuan: Sure, I am not sure either as to where it belongs. Most questions answered on SE are somewhat basic, and MO is very advanced, and I don't even know what this problem will turn out to be! – Bravo Mar 16 '12 at 18:21