Non-uniform Berry-Esseen bounds and p-Poincare inequalities in discrete setting

• Applicant: Professor Dr. Peter Eichelsbacher
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 547295909

The project mainly considers functionals of Rademacher random variables and applications which all can be considered as L^2-functionals of these random objects. Some of the examples are statistics in the context of random graphs, which can also be considered as counting statistics where there is no natural ordering of the summands. This is called a class of decomposable random variables due to Barbour, Karonski and Rucinski. On the road, we also study exchangeable pairs of random objects due to Charles Stein. Possible applications include statistics of the Erdös-Renyi random graph model like standardized subgraph counts or the number of vertices with prescribed degree. Other examples are infinitive weighted 2-runs and weighted U-statistics, the number of vertices of fixed degree studied for percolation on the Hamming hypercube and the number of isolated faces in the Linial-Meshulam-Wallach random kappa-complex. Our main goal is to study non-uniform Berry-Esseen bounds and refinements of Poincare inequalities for our functionals. We are planing to use and to develop further the method of Stein and the method of Malliavin-Stein to consider non-uniform bounds as well as the Stein-Tikhomirov method, where Stein's method is combined with the theory of characteristic functions to derive Berry-Esseen bounds in a setting of weakly dependent random variables. Finally, we will develop new explicit bounds on the Gaussian approximation of Rademacher functionals based on novel estimates of moments of Skorohod integrals to obtain improved Poincare-inequalities. The idea is to be able to derive bounds in the Wasserstein-distance and in the Kolmogorov-distances whose application requires minimal moment assumptions. Summarizing our goals comprise fine asymptotics under minimal moment assumptions.

• DFG Programme: Research Grants
• International Connection: United Kingdom
• Cooperation Partners: Professorin Dr. Gesine Reinert; Dr. Tara Trauthwein, Ph.D.