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There are two ways to discretize: rounding and conditioning. The first discretization you use to define $D_{\mathbb Z,s}$ from $\rho_s$ is using conditionning. The second one defining $\bar \Psi_s$ from $\Psi_s$ is using rounding (the density at integer $x$ of the former distribution is given the total density of the interval $[x-\frac 1 2, x+ \frac 12]$ of ...