Normal approximation for sums of weighted U-statistics - application to Kolmogorov bounds in random subgraph counting

摘要

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 Erdos-Renyi 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.