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.

there's no such result regarding K-S statistics applied to different populations. generally, it may not even be possible to calculate K-S stat. consider $X\sim\mathcal N$ and $Y\sim\chi^2$ - these two have different domains. how do you suppose to obtain K-S stat for them?
