Kolmogorov distance ($d_K(F,G)={\sup }_z|F(z)-G(z)|$)

The Kolmogorov distance between two probability distributions with cumulative distribution functions $F,G$ is ${\sup }_{z\in \mathbb{R}}|F(z)-G(z)|$.

If a cumulative distribution function is constant between consecutive lattice points spaced by $h$ near zero, then its Kolmogorov distance from the standard normal distribution function is at least half the increase of that function across one such interval. This gives a lower bound of order $h$.