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Moderate deviations for weight-dependent random connection models

Part of: Stochastic processes Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  04 June 2025

Nils Heerten ,
Christian Hirsch Open the ORCID record for Christian Hirsch [Opens in a new window]  and
Moritz Otto Open the ORCID record for Moritz Otto [Opens in a new window]
Nils Heerten*
Affiliation:
Ruhr University Bochum
Christian Hirsch*
Affiliation:
Aarhus University
Moritz Otto*
Affiliation:
Leiden University
*
*Email address: nils.heerten@rub.de
**Email address: hirsch@math.au.dk
***Email address: m.f.p.otto@math.leidenuniv.nl
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Abstract

In this paper we derive cumulant bounds for subgraph counts and power-weighted edge lengths in a class of spatial random networks known as weight-dependent random connection models. These bounds give rise to different probabilistic results, from which we mainly focus on moderate deviations of the respective statistics, but also show a concentration inequality and a normal approximation result. This involves dealing with long-range spatial correlations induced by the profile function and the weight distribution. We start by deriving the bounds for the classical case of a Poisson vertex set, and then provide extensions to α-determinantal processes.

Keywords

Moderate deviationsα-determinantal process

MSC classification

Primary: 60F10: Large deviations
Secondary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 4 , December 2025 , pp. 1406 - 1428
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Błaszczyszyn, B., Yogeshwaran, D. and Yukich, J. E. (2019). Limit theory for geometric statistics of point processes having fast decay of correlations. Ann. Prob. 47, 835–895.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Döring, H., Jansen, S. and Schubert, K. (2021). The method of cumulants for the normal approximation. Prob. Surveys. 19, 185–270.Google Scholar
Eichelsbacher, P., Raič, M. and Schreiber, T. (2015). Moderate deviations for stabilizing functionals in geometric probability. Ann. Inst. H. Poincaré Prob. Statist. 51, 89–128.CrossRefGoogle Scholar
Feng, R., Götze, F. and Yao, D. (2024). Determinantal point processes on spheres: Multivariate linear statistics. J. Funct. Anal. 287, 110493.CrossRefGoogle Scholar
Forrester, P. J. and Honner, G. (1999). Exact statistical properties of the zeros of complex random polynomials. J. Phys. A. 32, 2961–2981.CrossRefGoogle Scholar
Gracar, P., Grauer, A., Lüchtrath, L. and Mörters, P. (2019). The age-dependent random connection model. Queueing Systems 93, 309–331.CrossRefGoogle Scholar
Gracar, P., Heydenreich, M., Mönch, C. and Mörters, P. (2022). Recurrence versus transience for weight-dependent random connection models. Electron. J. Prob. 27, 60.CrossRefGoogle Scholar
Heinrich, L. (2016). On the strong Brillinger-mixing property of $\alpha$ -determinantal point processes and some applications. Appl. Math. 61, 443–461.CrossRefGoogle Scholar
Heinrich, L. and Klein, S. (2014). Central limit theorems for empirical product densities of stationary point processes. Stat. Inference Stoch. Process. 17, 121–138.CrossRefGoogle Scholar
Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. American Mathematical Society, Providence.CrossRefGoogle Scholar
Jolivet, E. (1981). Central limit theorem and convergence of empirical processes for stationary point processes. In Point Processes and Queuing Problems. North-Holland, Amsterdam & New York.Google Scholar
Komjáthy, J. and Lodewijks, B. (2020). Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs. Stoch. Process. Appl. 130, 1309–1367.CrossRefGoogle Scholar
Last, G. and Penrose, M. D. (2018). Lectures on the Poisson Process. Cambridge University Press, Cambridge.Google Scholar
Liu, Q. and Privault, N. (2024). Normal approximation of subgraph counts in the random-connection model. Bernoulli 30, 3224–3250.CrossRefGoogle Scholar
Penrose, M. D. (2003). Random Geometric Graphs. Oxford University Press, Oxford.CrossRefGoogle Scholar
Penrose, M. D. (2018). Inhomogeneous random graphs, isolated vertices, and Poisson approximation. J. Appl. Prob. 55, 112–136.CrossRefGoogle Scholar
Rudzkis, R., Saulis, L. and Statulevičius, V. (1978). A general lemma on probabilities of large deviations. Litovsk. Mat. Sb. 18, 99–116, 217.Google Scholar
Saulis, L. and Statulevičius, V. A. (1991). Limit Theorems for Large Deviations. Kluwer Academic, Dordrecht.CrossRefGoogle Scholar
Schulte, M. and Thäle, C. (2024). Moderate deviations on Poisson chaos. Electron. J. Prob. 29, 146.CrossRefGoogle Scholar
Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants I: Fermion, Poisson and Boson point processes. J. Funct. Anal. 205, 414–463.CrossRefGoogle Scholar
Statulevičius, V. (1966). On large deviations. Z. Wahrscheinlichkeitsth. 6, 133–144.CrossRefGoogle Scholar
Yogeshwaran, D. and Adler, R. (2015). On the topology of random complexes built over stationary point processes. Ann. Appl. Prob. 15, 3338–3380.Google Scholar