Say that a random variable $X$ follows the Gaussian distribution $\mathcal{N}(\mu, \sigma)$. Then will the ceiling $\lceil X\rceil$, the floor $\lfloor X\rfloor$, and the rounding $\lfloor X\rceil$ follow some discrete Gaussian distributions? Could you prove or disprove it? And if they do follow, what are the means and the standard deviations?

This type of discretized normal distribution occurs in some practical problems. For example, the median of the vertex degrees of an ErdősRenyi random graph with fixed edge probability has such a distribution with bounded variance. (This is a result of myself and Wormald that we still haven't published more than 10 years later.) I don't recall having seen a name for this distribution, and I don't know of a useful expression for its mean or variance. Any information about those things would be appreciated. Regarding the question, if $\sigma\to\infty$ the distribution is asymptotically normal in some senses (such as total variation distance), while if $\sigma=O(1)$ it isn't. 


I don't know what you mean by "discrete Gaussian distribution". Explicitly, the probability mass function of $\lceil X \rceil$ is $P(\lceil X \rceil = k) = P(k1 < X \le k) = F(k)  F(k1)$, where $F$ is the cumulative distribution function of $X$, and similarly the pmf's of $\lfloor X \rfloor$ and $\lfloor X \rceil$ can be written in terms of $F$. 

