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.

## References

M.Abramowitz and I.A.Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1964. 0171.38503M.Abramowitz and I.A.Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1964. 0171.38503

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. 1014.60025 10.1214/aoap/1037125857 euclid.aoap/1037125857Z.-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. 1014.60025 10.1214/aoap/1037125857 euclid.aoap/1037125857

J.H.Curtiss. A note on the theory of moment generating functions, Ann. of Math. Statist 13 (4) (1942), 430–433. 0063.01024 10.1214/aoms/1177731541 euclid.aoms/1177731541J.H.Curtiss. A note on the theory of moment generating functions, Ann. of Math. Statist 13 (4) (1942), 430–433. 0063.01024 10.1214/aoms/1177731541 euclid.aoms/1177731541

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. 1193.60011P.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. 1193.60011

P.Flajolet, J.Gabarro and H. Pekari, Analytic urns, Annals of Probability 33, (2005), 1200-1233. MR2135318 1073.60007 10.1214/009117905000000026 euclid.aop/1115386724P.Flajolet, J.Gabarro and H. Pekari, Analytic urns, Annals of Probability 33, (2005), 1200-1233. MR2135318 1073.60007 10.1214/009117905000000026 euclid.aop/1115386724

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. 1358.60014P.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. 1358.60014

R.L.Graham, D.E.Knuth and O.Patashnik, Concrete mathematics. Second Edition, Addison-Wesley Publishing Company, Reading, MA, 1994. 0836.00001R.L.Graham, D.E.Knuth and O.Patashnik, Concrete mathematics. Second Edition, Addison-Wesley Publishing Company, Reading, MA, 1994. 0836.00001

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.pdfH.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

S.Janson, Functional limit theorems for multitype branching processes and generalized Polya urns, Stochastic processes and applications. 110 (2004), 177-245. 1075.60109 10.1016/j.spa.2003.12.002S.Janson, Functional limit theorems for multitype branching processes and generalized Polya urns, Stochastic processes and applications. 110 (2004), 177-245. 1075.60109 10.1016/j.spa.2003.12.002

S.Janson, Limit theorems for triangular urn schemes, Probability Theory and Related Fields. 134 (2005), 417-452. 1112.60012 10.1007/s00440-005-0442-7S.Janson, Limit theorems for triangular urn schemes, Probability Theory and Related Fields. 134 (2005), 417-452. 1112.60012 10.1007/s00440-005-0442-7

N.L.Johnson and S.Kotz, Urn models and their application. An approach to modern discrete probability theory. John Wiley, New York, 1977. 0352.60001N.L.Johnson and S.Kotz, Urn models and their application. An approach to modern discrete probability theory. John Wiley, New York, 1977. 0352.60001

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. 0888.60014S.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. 0888.60014

B.Pittel, An urn model for cannibal behavior, Journal of Applied Probability 24 (1987), 522-526. 0637.60017 10.2307/3214275B.Pittel, An urn model for cannibal behavior, Journal of Applied Probability 24 (1987), 522-526. 0637.60017 10.2307/3214275

B.Pittel, On a Daley-Kendall Model of Random Rumours, Journal of Applied Probability 27 (1) (1990), 14-27. MR1039181 0698.60061 10.2307/3214592B.Pittel, On a Daley-Kendall Model of Random Rumours, Journal of Applied Probability 27 (1) (1990), 14-27. MR1039181 0698.60061 10.2307/3214592

B.Pittel, Normal convergence problem? Two moments and a recurrence may be the clues. Ann. Appl. Probab 9 (4) (1999), 1260-1302. 0960.60014 10.1214/aoap/1029962872 euclid.aoap/1029962872B.Pittel, Normal convergence problem? Two moments and a recurrence may be the clues. Ann. Appl. Probab 9 (4) (1999), 1260-1302. 0960.60014 10.1214/aoap/1029962872 euclid.aoap/1029962872

N.Pouyanne, Classification of large Pólya-Eggenberger urns with regard to their asymptotics. Discrete Mathematics and Theoretical Computer Science AD (2005), 177-245. MR2193125N.Pouyanne, Classification of large Pólya-Eggenberger urns with regard to their asymptotics. Discrete Mathematics and Theoretical Computer Science AD (2005), 177-245. MR2193125