Electronic Communications in Probability

Analysis of a class of Cannibal urns

Markus Kuba

Full-text: Open access

Abstract

In this note we study a class of $2\times 2$ Polya-Eggenberger urn models, which serves as a stochastic model in biology describing cannibalistic behavior of populations. A special case was studied before by Pittel using asymptotic approximation techniques, and more recently by Hwang et al. using generating functions. We obtain limit laws for the stated class of so-called cannibal urns by using Pittel's method, and also different techniques.

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 51, 583-599.

Dates
Accepted: 3 August 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465262007

Digital Object Identifier
doi:10.1214/ECP.v16-1669

Mathematical Reviews number (MathSciNet)
MR2846652

Zentralblatt MATH identifier
1245.60029

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 05C05: Trees

Keywords
Cannibal Urn models Normal distribution Poisson distribution

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kuba, Markus. Analysis of a class of Cannibal urns. Electron. Commun. Probab. 16 (2011), paper no. 51, 583--599. doi:10.1214/ECP.v16-1669. https://projecteuclid.org/euclid.ecp/1465262007


Export citation

References

  • M.Abramowitz and I.A.Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1964.
  • Z.-D.Bai, F.Hu and L.-X. Zhang. Gaussian approximation theorems for urn models and their applications, Annals of applied probability 12 (4) (2001), 1149-1173.
  • A.D.Barbour, L.Holst and S.Janson, Poisson Approximation. Oxford University Press, Oxford, UK, 1992.
  • J.H.Curtiss. A note on the theory of moment generating functions, Ann. of Math. Statist 13 (4) (1942), 430–433.
  • P.Dumas, P.Flajolet and V.Puyhaubert. Some exactly solvable models of urn process theory. Discrete Mathematics and Computer Science, Proceedings of Fourth Colloquium on Mathematics and Computer Science AG (2006), P. Chassaing Editor, 59-118.
  • P.Flajolet, J.Gabarro and H. Pekari, Analytic urns, Annals of Probability 33, (2005), 1200-1233.
  • P.Flajolet and T.Huillet, Analytic Combinatorics of the Mabinogion Urn. Discrete mathematics and Theoretical Computer Science AI (DMTCS), Proceedings of Fifth Colloquium on Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities (2008), U. Rösler editor, 549-572.
  • R.L.Graham, D.E.Knuth and O.Patashnik, Concrete mathematics. Second Edition, Addison-Wesley Publishing Company, Reading, MA, 1994.
  • H.K.Hwang, M.Kuba and A.Panholzer, Analysis of some exactly solvable diminishing urn models, in: Proceedings of the 19th International Conference on Formal Power Series and Algebraic Combinatorics, Nankai University, Tianjin (2007). Available at http://www.fpsac.cn/PDF-Proceedings/Posters/43.pdf
  • H.K.Hwang, M.Kuba and A.Panholzer, Diminishing urn models: Analysis of exactly solvable models, manuscript.
  • S.Janson, Functional limit theorems for multitype branching processes and generalized Polya urns, Stochastic processes and applications. 110 (2004), 177-245.
  • S.Janson, Limit theorems for triangular urn schemes, Probability Theory and Related Fields. 134 (2005), 417-452.
  • N.L.Johnson and S.Kotz, Urn models and their application. An approach to modern discrete probability theory. John Wiley, New York, 1977.
  • S.Kotz and N.Balakrishnan, Advances in urn models during the past two decades, in: Advances in combinatorial methods and applications to probability and statistics Stat. Ind. Technol., Birkhäuser, Boston (1997), 203-257.
  • H.Mahmoud, Urn models and connections to random trees: a review. Journal of the Iranian Mathematical Society 2 (2003), 53-114.
  • B.Pittel, An urn model for cannibal behavior, Journal of Applied Probability 24 (1987), 522-526.
  • B.Pittel, On tree census and the giant component in sparse random graphs, Random Structures & Algorithms 1, (1990), 311-332.
  • B.Pittel, On a Daley-Kendall Model of Random Rumours, Journal of Applied Probability 27 (1) (1990), 14-27.
  • B.Pittel, Normal convergence problem? Two moments and a recurrence may be the clues. Ann. Appl. Probab 9 (4) (1999), 1260-1302.
  • N.Pouyanne, Classification of large Pólya-Eggenberger urns with regard to their asymptotics. Discrete Mathematics and Theoretical Computer Science AD (2005), 177-245.
  • N.Pouyanne, An algebraic approach to Pólya processes. Annales de l'Institut Henri Poincaré 44 (2) (2008), 293-323.