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Short distances, flat triangles and Poisson limits

  • Bernard Silverman (a1) and Tim Brown (a2)

Abstract

Motivated by problems in the analysis of spatial data, we prove some general Poisson limit theorems for the U-statistics of Hoeffding (1948). The theorems are applied to tests of clustering or collinearities in plane data; nearest neighbour distances are also considered.

Copyright

Corresponding author

Now at the University of Bath.

References

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Brown, B. M. and Eagleson, G. K. (1971) Martingale convergence to infinitely divisible laws with finite variance. Trans. Amer. Math. Soc. 162, 449453.
Brown, T. C. (1978) A martingale approach to the Poisson convergence of point processes. Ann. Prob. 6, 615628.
Brown, T. C. and Silverman, B. W. (1979) Rates of Poisson convergence for U-statistics, J. Appl. Prob. To appear.
Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293325.
Kallenberg, O. (1973) Characterisation and convergence of random measures and point processes. Z. Wahrscheinlichkeitsth. 27, 921.
Kaplan, N. (1977) Two applications of a Poisson approximation for dependent events. Ann. Prob. 5, 787794.
Ripley, B. D. (1977) Modelling spatial patterns. J. R. Statist. Soc. B 39, 172212.
Ripley, B. D. and Silverman, B. W. (1978) Quick tests for spatial interaction. Biometrika 65.
Saunders, R. and Funk, G. M. (1977) Poisson limits for a clustering model of Strauss. J. Appl. Prob. 14, 776784.
Silverman, B. W. (1976) Limit theorems for dissociated random variables. Adv. Appl. Prob. 8, 806819.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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