# Discretization of a continuous distribution

For a research project I work with continuous distributions, like the normal distribution. In my use case however the random variable Z generally follows a normal distribution, though it can only take discrete values (..., -3, -2, -1, 0, 1, 2, 3, ...)$\in \mathbb{Z}$.

For the "discretization" I use a simple heuristic. An easy example would be a standard normal distribution:

$$Z \sim \mathcal{N}(0,1) \\ f(Z) = \sum_{i\in \mathbb{Z}} \mathbb{1}_{(i-0.5, i+0.5]}(Z) \\ X = f(Z)$$

My question now is: Is there a special name for the resulting distribution X? Is there any research on the resulting distribution?

Best, Simon

• This is called the discretized normal distribution on p. 2 of arxiv.org/pdf/1111.3162.pdf . See the paper for various theoretical results. Jul 14 '16 at 0:17

## 1 Answer

In lattice cryptography, discrete gaussian distribution is used a lot. However, there, it is defined in a different way: $$\mathbb P(X = k) = \dfrac{1}{S} e^{-k^2/2}, \quad S = \sum_{k \in \mathbb Z} e^{-k^2 / 2}.$$ Probably, they use this distribution because it has the maximal entropy among discrete distributions with fixed variance. I refer to ''Compact and Side Channel Resistant Discrete Gaussian Sampling'' for applications in cryptography (they consider problems of sampling with Knuth-Yao algorithm, and some more computational issues). I also refer to ''Probability distributions and maximum entropy'' for principle of maximum entropy - I didn't find the proof of maximal entropy for discrete gaussian mentioned above, but probably using the tools described in the article, you can do that.

What you use here, is rather $\lfloor \mathcal N(0,1) + \tfrac12 \rfloor$ with probability 1 (I mean, deciding on the value at the half-integers).