Random Environments: Cox Point Processes

  • Benedikt Jahnel
  • Wolfgang König
Part of the Compact Textbooks in Mathematics book series (CTM)


Modeling a system of telecommunication devices in space via a homogeneous PPP represents a situation where no information about the environment or any preferred behavior of the devices is available. To some degree this can be compensated by the use of a non-homogeneous PPP with general intensity measure μ, where areas can be equipped with higher or lower device density. Thereby we leave the mathematically nicer setting of spatial stationarity, but at least we keep the spatial independence. Nevertheless, the independence of devices is an assumption that is often violated in the real world since user behavior is usually correlated. One way to incorporate dependencies into the distribution of devices in space is to use Cox point processes, which are PPPs with a random intensity measure representing the environment. In the simplest case, the scalar intensity λ > 0 of a homogeneous PPP can now be seen as a random variable Λ taking values in [0, ) representing, for example temporal fluctuations in the intensity of devices in a city. The joint distribution remains stationary, but it is highly spatially correlated and in particular not ergodic, see  Chap. 6.


  1. [BolRio08]
    B. Bollobás and O. Riordan, Percolation on random Johnson–Mehl tessellations and related models, Probab. Theory Relat. Fields140:3–4, 319–343 (2008).Google Scholar
  2. [CHJ19]
    E. Cali, C. Hirsch and B. Jahnel, Continuum percolation for Cox processes, Stoch. Process. Their Appl.129:10, 3941–3966 (2019).Google Scholar
  3. [Cou12]
    T. Courtat, Promenade dans les cartes de villes-phénoménologie mathématique et physique de la ville-une approche géométrique, Ph.D. dissertation, Université Paris-Diderot, Paris (2012).Google Scholar
  4. [CKMS13]
    S.N. Chiu, W.S. Kendall, J. Mecke and D. Stoyan, Stochastic Geometry and Its Applications, J. Wiley & Sons, Chichester (2013).CrossRefGoogle Scholar
  5. [DalVer03]
    D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods, Second Edition, Springer (2003).zbMATHGoogle Scholar
  6. [DalVer08]
    D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, Second Edition, Springer (2008).CrossRefGoogle Scholar
  7. [Hän12]
    M. Hänggi, Stochastic Geometry for Wireless Networks, Cambridge University Press (2012).Google Scholar
  8. [JahTob19]
    B. Jahnel and A. Tóbiás, Exponential moments for planar tessellations, J. Stat. Phys.179, 90–109 (2020).MathSciNetCrossRefGoogle Scholar
  9. [vLie12]
    M. van Lieshout, An introduction to planar random tessellation models, Spat. Stat.E76, 40–49 (2012).CrossRefGoogle Scholar
  10. [LauZuy08]
    C. Lautensack and S. Zuyev, Random Laguerre tessellations, Adv. in Appl. Probab.40:3, 630–650 (2008).Google Scholar
  11. [Møl12]
    J. Møller, Lectures on Random Voronoi Tessellations, Springer (2012).Google Scholar
  12. [MølSto07]
    J. Møller and D. Stoyan, Stochastic Geometry and Random Tessellations, Tech. rep.: Department of Mathematical Sciences, Aalborg University (2007).Google Scholar
  13. [OBSC00]
    A. Okabe, B. Boots, K. Sugihara and S.N. Chui, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, J. Wiley & Sons, Chichester (2000).CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Benedikt Jahnel
    • 1
  • Wolfgang König
    • 1
    • 2
  1. 1.Weierstraß-InstitutBerlinGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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