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Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields

Part of: Multivariate analysis Inference from stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  14 July 2023

Satoshi Kuriki  and
Takahiko Matsubara
Satoshi Kuriki*
Affiliation:
The Institute of Statistical Mathematics
Takahiko Matsubara*
Affiliation:
High Energy Accelerator Research Organization (KEK)
*
*Postal address: The Institute of Statistical Mathematics, 10-3 Midoricho, Tachikawa, Tokyo 190-8562, Japan. Email address: kuriki@ism.ac.jp
**Postal address: Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan. Email address: tmats@post.kek.jp
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Abstract

The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, whose k-point correlation functions (kth-order cumulants) have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-squared random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.

Keywords

Chi-squared random fieldEuler characteristic densityGaussian-related random fieldsk-point correlation function

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62M40: Random fields; image analysis 62H10: Distribution of statistics
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 4 , December 2023 , pp. 1390 - 1414
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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