arXiv subjects

Benedikt Rednoß

Publications and source records attributed to Benedikt Rednoß.

6 recordsLinked to original sources

arXiv2026

Central limit theorems for high dimensional lattice polytopes: cosmological polytopes

We study cosmological polytopes induced by Erdős--Rényi random graphs in a high-dimensional regime. These graph-based lattice polytopes form a natural model of random lattice polytopes in which geometric features are determined by the structure of the underlying random graph. Focusing on the number of polytope edges and on the number of edges in unimodular triangulations, we derive asymptotic formulas for expectations and variances and prove quantitative central limit theorems in the relevant parameter regime. The analysis relies on explicit graph-theoretic descriptions of the corresponding edge sets and on the discrete Malliavin--Stein method for normal approximation.

math.CO↗

arXiv2026

Central limit theorems for high dimensional lattice polytopes: symmetric edge polytopes

We investigate symmetric edge polytopes generated by Erdős--Rényi random graphs in a high-dimensional regime. These objects provide a natural and largely unexplored model of random lattice polytopes, in which geometric properties are governed by graph-theoretic structure. Focusing on the number of polytope edges and on the number of edges in unimodular triangulations, we derive precise asymptotics for expectations and variances and establish central limit theorems with explicit rates of convergence. Our analysis combines a detailed combinatorial-geometric study of the graph configurations determining the facial structure with the discrete Malliavin--Stein method for normal approximation. In particular, we identify a distinguished parameter value at which the leading variance term cancels, producing an atypical fluctuation regime. To the best of our knowledge, the results obtained here constitute the first distributional limit theorems for random lattice polytopes

math.CO↗

arXiv2024

Moderate Deviations for Functionals over infinitely many Rademacher random variables

In this paper, moderate deviations for normal approximation of functionals over infinitely many Rademacher random variables are derived. They are based on a bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, continued by an intensive study of the behavior of operators from the Malliavin--Stein method along with the moment generating function of the mentioned functional. As applications, subgraph counting in the Erdős--Rényi random graph and infinite weighted 2-runs are studied.

math.PR↗

arXiv2024

Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions

This paper deals with sequences of random variables $X_n$ only taking values in $\{0,\ldots,n\}$. The probability generating functions of such random variables are polynomials of degree $n$. Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for $X_n$ is established in a unified way. In the real rooted case the result is classical and only involves the variances of $X_n$, while in the cyclotomic case the fourth cumulants or moments of $X_n$ appear in addition. The proofs are elementary and based on the Stein-Tikhomirov method.

math.PR↗

arXiv2021

A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

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 Erdös-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.

math.PR↗

arXiv2021

Kolmogorov bounds for decomposable random variables and subgraph counting by the Stein-Tikhomirov method

In his work \cite{Ti80}, Tikhomirov combined elements of Stein's method with the theory of characteristic functions to derive Kolmogorov bounds for the convergence rate in the central limit theorem for a normalized sum of a stationary sequence of random variables satisfying one of several weak dependency conditions. The combination of elements of Stein's method with the theory of characteristic functions is sometimes called \emph{Stein-Tikhomirov method}. \citet*{AMPS17} successfully used the Stein-Tikhomirov method to bound the convergence rate in contexts with non-Gaussian targets. \citet*{Ro17} used the Stein-Tikhomirov method to bound the convergence rate in the Kolmogorov distance for normal approximation of normalized triangle counts in the Erdös-Rényi random graph.

math.PR↗