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Self-crossing Points of a Line Segment Process

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  • Zbyněk Pawlas

    (Charles University in Prague)

Abstract

This paper is devoted to planar stationary line segment processes. The segments are assumed to be independent, identically distributed, and independent of the locations (reference points). We consider a point process formed by self-crossing points between the line segments. Its asymptotic variance is explicitly expressed for Poisson segment processes. The main result of the paper is the central limit theorem for the number of intersection points in expanding rectangular sampling window. It holds not only for Poisson processes of reference points but also for stationary point processes satisfying certain conditions on absolute regularity (β-mixing) coefficients. The proof is based on the central limit theorem for β-mixing random fields. Approximate confidence intervals for the intensity of intersections can be constructed.

Suggested Citation

  • Zbyněk Pawlas, 2014. "Self-crossing Points of a Line Segment Process," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 295-309, June.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:2:d:10.1007_s11009-012-9315-6
    DOI: 10.1007/s11009-012-9315-6
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    References listed on IDEAS

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    1. Lothar Heinrich & Michaela Prokešová, 2010. "On Estimating the Asymptotic Variance of Stationary Point Processes," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 451-471, September.
    2. Pawlas, Zbynek & Honzl, Ondrej, 2010. "Comparison of length-intensity estimators for segment processes," Statistics & Probability Letters, Elsevier, vol. 80(9-10), pages 825-833, May.
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    Cited by:

    1. Jakub Večeřa & Viktor Beneš, 2016. "Interaction Processes for Unions of Facets, the Asymptotic Behaviour with Increasing Intensity," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 1217-1239, December.
    2. Viktor Beneš & Jakub Večeřa & Milan Pultar, 2019. "Planar Segment Processes with Reference Mark Distributions, Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 683-698, September.

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