Limit theorems for generalized perimeters of random inscribed polygons. I

Authors

  • Ekaterina N. Simarova St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation, Leonhard Euler International Mathematical Institute, 29B, 14 liniya V. O., St. Petersburg, 199178, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2020.409

Abstract

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.

Keywords:

U-max-statistics, Poisson approximation, generalized perimeter, limiting behavior

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References

Литература

1. Lao W. Some weak limit laws for the diameter of random point sets in bounded regions. Ph.D. Thesis. Karlsruhe, 2010.

2. Mayer M. Random Diameters and Other U-max-Statistics. Ph.D. Thesis. Bern University, 2008.

3. Barbour A. D., Holst L., Janson S. Poisson Approximation. London: Oxford University Press, 1992.

4. Lao W., Mayer M. U-max-statistics // J. Multivariate Anal. 2008. Vol. 99. P. 2039–2052.

5. Silverman F. B., Brown T. Short distances, flat triangles, and Poisson limits // J. Appl. Probab. 1978. Vol. 15. P. 815–825.

6. Koroleva E. V., Nikitin Ya. Yu. U-max-statistics and limit theorems for perimeters and areas of random polygons // J. Multivariate Anal. 2014. Vol. 127. P. 99–111.

7. Yaglom I. M., Boltyanskii V. G. Convex figures. Transl. by P. J. Kelly and L. F. Walton. New York: Holt, Rinehart and Winston, 1961.

8. Legendre A. M. Elements of Geometry and Trigonometry: With Notes. Oliver & Boyd, 1822.

References

1. Lao W., Some weak limit laws for the diameter of random point sets in bounded regions (Ph.D. Thesis, Karlsruhe, 2010).

2. Mayer M., Random Diameters and Other U-max-Statistics (Ph.D. Thesis, Bern University, 2008).

3. Barbour A. D., Holst L., Janson S., Poisson Approximation (Oxford University Press, London, 1992).

4. Lao W., Mayer M., “U-max-statistics”, J. Multivariate Anal. 99, 2039–2052 (2008).

5. Silverman F. B., Brown T., “Short distances, flat triangles, and Poisson limits”, J. Appl. Probab.15, 815–825 (1978).

6. Koroleva E. V., Nikitin Ya. Yu., “U-max-statistics and limit theorems for perimeters and areas of random polygons”, J. Multivariate Anal. 127, 99–111 (2014).

7. Yaglom I. M., Boltyanskii V. G., Convex figures (Transl. by P. J. Kelly and L. F. Walton, Holt, Rinehart and Winston, New York, 1961).

8. Legendre A. M., Elements of Geometry and Trigonometry: With Notes (Oliver & Boyd, 1822).

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Published

2020-12-27

How to Cite

Simarova, E. N. (2020). Limit theorems for generalized perimeters of random inscribed polygons. I. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(4), 678–687. https://doi.org/10.21638/spbu01.2020.409

Issue

Vol. 7 No. 4 (2020)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.