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Transforming spatial point processes into Poisson processes

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  • Schoenberg, Frederic

Abstract

In 1986, Merzbach and Nualart demonstrated a method of transforming a two-parameter point process into a planar Poisson process of unit rate, using random stopping sets. Merzbach and Nualart's theorem applies only to a special class of point processes, since it requires two restrictive conditions: (F4) condition of conditional independence and the convexity of the 1-compensator. (F4) condition was removed in 1990 by Nair, but the convexity condition remained. Here both (F4) condition and the convexity condition are removed by making use of predictable sets rather than stopping sets. As with Nair's theorem, the result extends to point processes in higher dimensions.

Suggested Citation

  • Schoenberg, Frederic, 1999. "Transforming spatial point processes into Poisson processes," Stochastic Processes and their Applications, Elsevier, vol. 81(2), pages 155-164, June.
    • Handle: RePEc:eee:spapps:v:81:y:1999:i:2:p:155-164
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    References listed on IDEAS

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    1. P. Frevert, 1971. "Note," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 38(2), pages 269-270.
    2. Brown, Timothy C. & Ivanoff, B. G. & Weber, N. C., 1986. "Poisson convergence in two dimensions with application to row and column exchangeable arrays," Stochastic Processes and their Applications, Elsevier, vol. 23(2), pages 307-318, December.
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    Cited by:

    1. Nicoletta D’Angelo & Marianna Siino & Antonino D’Alessandro & Giada Adelfio, 2022. "Local spatial log-Gaussian Cox processes for seismic data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(4), pages 633-671, December.
    2. Peng, Roger, 2003. "Multi-dimensional Point Process Models in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 8(i16).
    3. Frederic Paik Schoenberg, 2004. "Testing Separability in Spatial-Temporal Marked Point Processes," Biometrics, The International Biometric Society, vol. 60(2), pages 471-481, June.
    4. Renato Assunção & Alexandra Maia, 2007. "A Note on Testing Separability in Spatial-Temporal Marked Point Processes," Biometrics, The International Biometric Society, vol. 63(1), pages 290-294, March.
    5. repec:jss:jstsof:08:i16 is not listed on IDEAS
    6. Giada Adelfio & Frederic Schoenberg, 2009. "Point process diagnostics based on weighted second-order statistics and their asymptotic properties," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 929-948, December.

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