Kolmogorov–Smirnov statistic/distance is normally distributed? Suppose we have two populations X and Y, then compute the KS statistic/distance for samples drawn from them. Is this KS statistic/distance normally distributed? If so, how to prove it? If not, what distribution should it be? I know samples means $\bar{X}$ and $\bar{Y}$ are normally distributed according to the central limit theory.

I'm assuming that you're talking about a case where X and Y are i.i.d. As noted, the distribution of the test statistic is not normal. Instead, it follows the Kologormov distribution. See the wikipedia entry: https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Kolmogorov_distribution
