Is there any literature discussing PDF after quantization?
any discrete random variable has a p.d.f. that is a summation of dirac delta functions.

$$ p_\mathrm{y}(\alpha) = \sum\limits_i P_i \ \delta(\alpha - y_i) $$

where $\sum\limits_i P_i = 1 $.

if $y[n]$ is the quantization of $x[n]$:

$$ y[n] = \Delta \bigg\lfloor \frac{x[n]}{\Delta} \bigg\rfloor $$

where $ \lfloor \cdot \rfloor$ is the floor() function and $\Delta$ is the quantization step size.

then $y_i = i \Delta $ for some set of integers $i$.  the p.d.f. weighting constants become

$$ P_i = \int\limits_{i \Delta}^{(i+1)\Delta} p_\mathrm{x}(\alpha) \, d \alpha $$