Asymptotic distribution of fiber intersection counts in two-dimensional random fiber webs

Heuiju Chun; Jae-Hwan Jhong; Heuiju Chun; Jae-Hwan Jhong

doi:10.3934/math.2026617

AIMS Mathematics

2026, Volume 11, Issue 5: 15008-15027. doi: 10.3934/math.2026617

Previous Article Next Article

Research article Topical Sections

Asymptotic distribution of fiber intersection counts in two-dimensional random fiber webs

Heuiju Chun ¹,
Jae-Hwan Jhong ^{2
,
,}

1.
Department of Statistics and Information, Dongduk Women's University, 60, Hwarang-ro 13-gil, Seongbuk-gu, Seoul, 02748, Republic of Korea
2.
Department of Information Statistics, Chungbuk National University, 1, Chungdae-ro, Cheongju-si, Chungcheongbuk-do, 28644, Republic of Korea

Received: 09 February 2026 Revised: 04 May 2026 Accepted: 18 May 2026 Published: 28 May 2026
MSC : 60D05, 62E20, 65C05

We study the asymptotic distribution of the number of fiber intersections in two-dimensional random fiber webs. Previous work on such systems has focused primarily on first- and second-order moment characteristics derived from geometric probability models. In this paper, we move beyond moment-based analysis and establish asymptotic normality for normalized intersection counts. For a planar fiber web consisting of line segments randomly placed with uniform locations and orientations, we show that the appropriately normalized number of intersections within a bounded observation window converges in distribution to a Gaussian limit as the number of fibers diverges. We first derive a central limit theorem for the equal-length fiber model and then extend the result to heterogeneous fiber systems with a mixture of two fiber lengths. In both cases, explicit expressions for the asymptotic mean and variance are obtained in terms of the first two moments of the truncated fiber lengths within the counting region. Theoretical results are supported by Monte Carlo simulations, which demonstrate that the Gaussian approximation provides an accurate description of the finite-sample distribution of the intersection count. Our findings place random fiber intersection models within the broader framework of stochastic geometry and provide a foundation for statistical inference in fiber-based network systems.
- asymptotic normality,
- random fiber web,
- geometric probability,
- stochastic geometry,
- intersection count
Citation: Heuiju Chun, Jae-Hwan Jhong. Asymptotic distribution of fiber intersection counts in two-dimensional random fiber webs[J]. AIMS Mathematics, 2026, 11(5): 15008-15027. doi: 10.3934/math.2026617

Related Papers:

Abstract

We study the asymptotic distribution of the number of fiber intersections in two-dimensional random fiber webs. Previous work on such systems has focused primarily on first- and second-order moment characteristics derived from geometric probability models. In this paper, we move beyond moment-based analysis and establish asymptotic normality for normalized intersection counts. For a planar fiber web consisting of line segments randomly placed with uniform locations and orientations, we show that the appropriately normalized number of intersections within a bounded observation window converges in distribution to a Gaussian limit as the number of fibers diverges. We first derive a central limit theorem for the equal-length fiber model and then extend the result to heterogeneous fiber systems with a mixture of two fiber lengths. In both cases, explicit expressions for the asymptotic mean and variance are obtained in terms of the first two moments of the truncated fiber lengths within the counting region. Theoretical results are supported by Monte Carlo simulations, which demonstrate that the Gaussian approximation provides an accurate description of the finite-sample distribution of the intersection count. Our findings place random fiber intersection models within the broader framework of stochastic geometry and provide a foundation for statistical inference in fiber-based network systems.

References

[1]	O. Kallmes, H. Corte, The structure of paper. Ⅰ. The statistical geometry of an ideal two dimensional fiber network, Tappi J., 43 (1960), 737–752.
[2]	H. W. Piekaar, L. A. Clarenburg, Aerosol filters—Pore size distribution in fibrous filters, Chem. Eng. Sci., 22 (1967), 1399–1407. https://doi.org/10.1016/0009-2509(67)80068-X doi: 10.1016/0009-2509(67)80068-X
[3]	A. Rawal, S. Lomov, T. Ngo, I. Verpoest, J. Vankerrebrouck, Mechanical behavior of thru-air bonded nonwoven structures, Text. Res. J., 77 (2007), 417–431. https://doi.org/10.1177/0040517507081313 doi: 10.1177/0040517507081313
[4]	T. Komori, Y. Ujihara, Y. Matsunaga, K. Makishima, Crossings of curled fibers in two-dimensional assemblies, Tappi, 62 (1979), 93–95.
[5]	Y. B. Yi, L. Berhan, A. M. Sastry, Statistical geometry of random fibrous networks, revisited: Waviness, dimensionality, and percolation, J. Appl. Phys., 96 (2004), 1318–1327. https://doi.org/10.1063/1.1763240 doi: 10.1063/1.1763240
[6]	J. H. Williams, S. S. Lee, C. S. Wasserman, Fiber intersections in a planar randomly oriented fiber composite, Fibre Sci. Technol., 10 (1977), 161–177. https://doi.org/10.1016/0015-0568(77)90018-5 doi: 10.1016/0015-0568(77)90018-5
[7]	M. W. Suh, H. Chun, R. L. Berger, P. Bloomfield, Distribution of fiber intersections in two-dimensional random fiber webs – A basic geometrical probability model, Text. Res. J., 80 (2010), 301–311. https://doi.org/10.1177/0040517509105071 doi: 10.1177/0040517509105071
[8]	H. Chun, Probabilistic modeling of fiber length segments within a bounded area of two-dimensional fiber webs, Commun. Stat. Appl. Methods, 18 (2011), 301–317.
[9]	H. Chun, M. W. Suh, Distribution of fiber intersections in two-dimensional random fiber web cases with a mixture of two fiber lengths, Text. Res. J., 90 (2020), 1851–1859. https://doi.org/10.1177/0040517519898158 doi: 10.1177/0040517519898158
[10]	M. Reitzner, M. Schulte, Central limit theorems for $u$-statistics of poisson point processes, Ann. Probab., 41 (2013), 3879–3909. https://doi.org/10.1214/12-AOP817 doi: 10.1214/12-AOP817
[11]	M. D. Penrose, J. E. Yukich, Central limit theorems for some graphs in computational geometry, Ann. Appl. Probab., 11 (2001), 1005–1041.
[12]	M. G. Kendall, P. A. P. Moran, Geometrical probability, Hafner Publishing Company, 1963.
[13]	D. A. Klain, G. C. Rota, Introduction to geometric probability, Cambridge: Cambridge University Press, 1997.
[14]	H. Solomon, Geometric probability, 1978.
[15]	G. Last, M. D. Penrose, Poisson process Fock space representation, chaos expansion and covariance inequalities, Probab. Theory Relat. Fields, 150 (2011), 663–690. https://doi.org/10.1007/s00440-010-0288-5 doi: 10.1007/s00440-010-0288-5
[16]	G. Peccati, M. Reitzner, Stochastic analysis for Poisson point processes: Malliavin calculus, wiener-Itô chaos expansions and stochastic geometry, Springer, 2016.
[17]	W. Hoeffding, A class of statistics with asymptotically normal distribution, In: Breakthroughs in statistics, New York: Springer, 308–334. https://doi.org/10.1007/978-1-4612-0919-5_20
[18]	A. J. Lee, $U$-Statistics: Theory and practice, 1990. https://doi.org/10.1201/9780203734520
[19]	P. K. Sen, Almost sure convergence of generalized $U$-statistics, Ann. Probab., 5 (1977), 287–290.
[20]	L. Zhuang, Y. Ma, G. Fang, A. Xu, Modeling two-scale degradation with heterogeneity: A unified random-effects inverse Gaussian framework, IISE Trans., 2026. https://doi.org/10.1080/24725854.2026.2631614 doi: 10.1080/24725854.2026.2631614
[21]	Y. Wang, J. Li, Q. Zhou, W. Wang, A unified framework for complex survival data: accounting for clustering, cure fractions, and competing risks, BMC Med. Res. Methodol., 26 (2026), 105. https://doi.org/10.1186/s12874-026-02842-z doi: 10.1186/s12874-026-02842-z

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)