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

Eichelsbacher, Peter; Rednoß, Benedikt

doi:10.48550/arxiv.2107.03775

Cited by 3 publications

(3 citation statements)

References 15 publications

(55 reference statements)

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“…Translating this result for uncentered subgraph counts would yield an approximation by a function of a multivariate normal. Several univariate normal approximation theorems for subgraph counts are available; recent developments in this area include [28], which uses Malliavin calculus together with Stein's method, and [11], which uses the Stein-Tikhomirov method.…”

Section: Related Workmentioning

confidence: 99%

Multivariate Central Limit Theorems for Random Clique Complexes

Temčinas¹,

Nanda²,

Reinert³

2021

Preprint

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Acyclic partial matchings on simplicial complexes play an important role in topological data analysis by facilitating efficient computation of (persistent) homology groups. Here we describe probabilistic properties of critical simplex counts for such matchings on clique complexes of Bernoulli random graphs. In order to accomplish this goal, we generalise the notion of a dissociated sum to a multivariate setting and prove an abstract multivariate central limit theorem using Stein's method. As a consequence of this general result, we are able to extract central limit theorems not only for critical simplex counts, but also for generalised U -statistics (and hence for clique counts in Bernoulli random graphs) as well as simplex counts in the link of a fixed simplex in a random clique complex.

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Section: Related Workmentioning

confidence: 99%

Multivariate Central Limit Theorems for Random Clique Complexes

Temčinas¹,

Nanda²,

Reinert³

2021

Preprint

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“…Those results have been made more precise in [BKR89] by the derivation of convergence rates in the Wasserstein distance via Stein's method. They have also been strengthened in [KRT17] using the Kolmogorov distance in the case of triangle counts, and in [PS20] in the case of general subgraphs G. The case of triangles has also been treated in [Röl22] by the Stein-Tikhomirov method, which has been extended to general subgraphs in [ER21]. In [Kho08], the counts of line (X-model) and cycles (Y -model) in discrete Erdős-Rényi models have been analyzed via the asymptotic behavior of their cumulants.…”

Section: Introductionmentioning

confidence: 99%

Normal approximation of subgraph counts in the random-connection model

Li¹,

Privault²

2023

Preprint

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This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by Möbius inversion. Using the Statulevičius condition, we deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity. Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.

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“…In full generality this theorem has first been proven in [29,Theorem 4.2] using the normal approximation bound from [17,Proposition 4.1] and a multiplication formula for discrete multiple stochastic integrals (the additional assumption in [29] that the graph Γ has no isolated vertices is not necessary and can be removed as we explain at the beginning of the proof of Theorem 1.2 in Section 6). Recently in [10], two of us derived an alternative proof, combining the decomposition of [3] with elements of Stein's method and with the theory of characteristic functions, sometimes called Stein-Tikhomirov method. In the present paper we will provide yet another proof of this result, which is almost purely combinatorial and, as we think, conceptually easier than the one in [29].…”

Section: Introduction and Applicationsmentioning

confidence: 99%

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

Eichelsbacher¹,

Rednoß²,

Thäle³

et al. 2021

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

Cited by 3 publications

References 15 publications

Multivariate Central Limit Theorems for Random Clique Complexes

Multivariate Central Limit Theorems for Random Clique Complexes

Normal approximation of subgraph counts in the random-connection model

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

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