We derive normal approximation bounds in the Kolmogorov distance for random variables posessing decompositions of Barbour, Karoński, and Ruciński [2]. We highlight the example of normalized subgraph counts in the Erdös-Rényi random graph. We prove a bound by generalizing the argumentation of Röllin [14], who used the Stein-Tikhomirov method to prove a bound in the special case of normalized triangle counts. Our bounds match the best available Wasserstein-bounds.

In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdős-Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random κ-complex and infinite weighted 2-runs are treated.

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