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Tara Trauthwein

Publications and source records attributed to Tara Trauthwein.

6 recordsLinked to original sources
arXiv2026

Second-order Poincar\'e inequalities and localization on the Poisson space

Given a mean zero functional $F$ of a Poisson measure on a metric space, we apply the Malliavin-Stein method to establish sharpened second-order Poincar\'e inequalities for $F/\sqrt{\operatorname{Var} (F)}$ in terms of fourth moments of difference operators. The rates of normal approximation are expressed in the Kolmogorov and Wasserstein distances and require fewer error terms than corresponding previous results. When $F$ is expressible as a sum of score functions which are distributionally close to scores having short-range structure, then we deduce that $F/\sqrt{\operatorname{Var}(F)}$ satisfies Berry-Esseen bounds. The normal approximation criteria of the scores, here called bounded Lipschitz localization, are more general than stabilization criteria and allow for unbounded interactions of scores. This approach yields Berry-Esseen bounds for local U-statistics on metric measures spaces, localizing functionals on hyperbolic space, as well as for Poisson functionals in a space-time setting, with infinite time horizon, including statistics of spatial birth-growth models and Laguerre tessellations.

math.PR↗
arXiv2025

Triadic structures in multislice networks

Networks provide a popular representation of complex data. Often, different types of relational measurements are taken on the same subjects. Such data can be represented as a \textit{multislice network}, a collection of networks on the same set of nodes, with connections between the different layers to be determined. For the analysis of multislice networks, we take inspiration from the analysis of simple networks, for which small subgraphs (motifs) have proven to be useful; motifs are even seen as building blocks of complex networks. A particular instance of a motif is a triangle, and while triangle counts are well understood for simple network models such as Erdős-Rényi random graphs, with i.i.d. distributed edges, even for simple multislice network models little is known about triangle counts. Here we address this issue by extending the analysis of triadic structures to multislice Erdős-Rényi networks. Again taking inspiration from the analysis of sparse Erdős-Rényi random graphs, we show that the distribution of triangles across multiple layers in a multislice Erdős-Rényi network can be well approximated by an appropriate Poisson distribution. This theoretical result opens the door to statistical goodness of fit tests for multislice networks.

math.PR↗
arXiv2024

Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space

Consider a stationary Poisson process $η$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $η$ as follows. First, each point $x\inη$ is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until $x$ is contained in the convex hull of the points already connected to $x$. The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered.

math.PR↗
arXiv2024

Multivariate Second-Order $p$-Poincaré Inequalities

In this work, we discuss new bounds for the normal approximation of multivariate Poisson functionals under minimal moment assumptions. Such bounds require one to estimate moments of so-called add-one costs of the functional. Previous works required the estimation of $4^{\text{th}}$ moments, while our result only requires $(2 + ε)$-moments, based on recent improvements introduced by (Trauthwein 2022). As applications, we show quantitative CLTs for two multivariate functionals of the Gilbert, or random geometric, graph. These examples were out of range for previous methods.

math.PR↗
arXiv2022

Quantitative CLTs on the Poisson space via Skorohod estimates and $p$-Poincaré inequalities

We establish new explicit bounds on the Gaussian approximation of Poisson functionals based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin-Stein method, we derive bounds in the Wasserstein and Kolmogorov distances whose application requires minimal moment assumptions on add-one cost operators $\unicode{x2014}$ thereby extending the results from (Last, Peccati and Schulte, 2016). Our applications include a CLT for the Online Nearest Neighbour graph, whose validity was conjectured in (Wade, 2009; Penrose and Wade, 2009). We also apply our techniques to derive quantitative CLTs for edge functionals of the Gilbert graph, of the $k$-Nearest Neighbour graph and of the Radial Spanning Tree, both in cases where qualitative CLTs are known and unknown.

math.PR↗