Analysis of a class of Cannibal urns

Markus Kuba (Technische Universität Wien)

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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 583-599

Publication Date: August 3, 2011

DOI: 10.1214/ECP.v16-1669

References

  1. M.Abramowitz and I.A.Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1964. Math. Review 29 #4914
  2. 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. Math. Review 2003k:60062
  3. A.D.Barbour, L.Holst and S.Janson, Poisson Approximation. Oxford University Press, Oxford, UK, 1992. Math. Review 93g:60043
  4. J.H.Curtiss. A note on the theory of moment generating functions, Ann. of Math. Statist 13 (4) (1942), 430--433. Math. Review 4,163f
  5. 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. Math. Review 2011b:60030
  6. P.Flajolet, J.Gabarro and H.~Pekari, Analytic urns, Annals of Probability 33, (2005), 1200-1233. Math. Review 2006m:60014
  7. 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. Math. Review 2010i:60032
  8. R.L.Graham, D.E.Knuth and O.Patashnik, Concrete mathematics. Second Edition, Addison-Wesley Publishing Company, Reading, MA, 1994. Math. Review 97d:68003
  9. 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
  10. H.K.Hwang, M.Kuba and A.Panholzer, Diminishing urn models: Analysis of exactly solvable models, manuscript.
  11. S.Janson, Functional limit theorems for multitype branching processes and generalized Polya urns, Stochastic processes and applications. 110 (2004), 177-245. Math. Review 2005a:60134
  12. S.Janson, Limit theorems for triangular urn schemes, Probability Theory and Related Fields. 134 (2005), 417-452. Math. Review 2007b:60051
  13. N.L.Johnson and S.Kotz, Urn models and their application. An approach to modern discrete probability theory. John Wiley, New York, 1977.
  14. 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.
  15. H.Mahmoud, Urn models and connections to random trees: a review. Journal of the Iranian Mathematical Society 2 (2003), 53-114.
  16. B.Pittel, An urn model for cannibal behavior, Journal of Applied Probability 24 (1987), 522-526. Math. Review 88e:60017
  17. B.Pittel, On tree census and the giant component in sparse random graphs, Random Structures & Algorithms 1, (1990), 311-332. Math. Review 92f:05087
  18. B.Pittel, On a Daley-Kendall Model of Random Rumours, Journal of Applied Probability 27 (1) (1990), 14-27. Math. Review 91k:60033
  19. B.Pittel, Normal convergence problem? Two moments and a recurrence may be the clues. Ann. Appl. Probab 9 (4) (1999), 1260-1302. Math. Review 2001a:60009
  20. N.Pouyanne, Classification of large Pólya-Eggenberger urns with regard to their asymptotics. Discrete Mathematics and Theoretical Computer Science AD (2005), 177-245. Math. Review 2193125
  21. N.Pouyanne, An algebraic approach to Pólya processes. Annales de l'Institut Henri Poincaré 44 (2) (2008), 293-323. Math. Review 2010b:60086
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.