Skip to main content
SpringerLink
Log in

Limit Theorems for Generalized Perimeters of Random Inscribed Polygons. I

  • MATHEMATICS
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

Recently, W. Lao and M. Mayer (2008) developed U-max-statistics, where instead of averaging the values of the kernel over various subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to the distributions of extreme values. In this paper, we begin to consider the limit theorems for the generalized perimeter (the sum of side powers) of a random inscribed polygon and U-max-statistics related to it. We describe extreme values of the generalized perimeter and obtain limit theorems for the cases where the side powers involved in determining the generalized perimeter do not exceed 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. W. Lao, Some Weak Limit Laws for the Diameter of Random Point Sets in Bounded Regions, PhD Thesis (Institut für Stochastik, Karlsruhe, 2010).

  2. M. Mayer, Random Diameters and Other U-max-Statistics, PhD Thesis (Univ. of Bern, Bern, 2008).

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. I. M. Yaglom and V. G. Boltyanskii, Convex Figures (Holt, Rinehart and Winston, New York, 1961).

    MATH  Google Scholar 

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

    Google Scholar 

Download references

Funding

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1619).

Author information

Authors and Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    E. N. Simarova

  2. Euler International Mathematical Institute, 197022, St. Petersburg, Russia

    E. N. Simarova

Authors
  1. E. N. Simarova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. N. Simarova.

Additional information

Translated by L. Kartvelishvili

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simarova, E.N. Limit Theorems for Generalized Perimeters of Random Inscribed Polygons. I. Vestnik St.Petersb. Univ.Math. 53, 434–442 (2020). https://doi.org/10.1134/S1063454120040093

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454120040093

Keywords:

Search

Navigation