We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and U -statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraphs 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 and improving recent results derived for triangles as well as results using the Wasserstein distance.

Abstract. We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0, 1) killed at 1 and killed or reflected at 0.

We propose Mecke-Palm formula for multiple integrals with respect to the Poisson random measure and its intensity measure performed, or mixed, in an arbitrary order. We apply the formulas to mixed Lévy systems of Lévy processes and obtain moment formulas for mixed Poisson integrals.

We provide sharp two-sided estimates on the Dirichlet heat kernel k1(t, x, y) for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel k1(t, x, y) in terms of its global two-sided sharp estimates.