We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and weighted $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraph counts in the Erdős–Rényi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering recent results obtained for triangles and improving other bounds in the Wasserstein distance.
"Normal approximation for sums of weighted $U$-statistics – application to Kolmogorov bounds in random subgraph counting." Bernoulli 26 (1) 587 - 615, February 2020. https://doi.org/10.3150/19-BEJ1141