### On a memory game and preferential attachment graphs

#### Abstract

In their recent paper Velleman and Warrington (2013) analyzed the expected values of some of the parameters in a memory game; namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by-product of our methods, we obtain the joint asymptotic normality of the degree counts in the preferential attachment graphs. Furthermore, we obtain simpler proofs (although without rate of convergence) of some of the results of Peköz et al. (2014) on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. In order to prove that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multitype generalized Pólya urn model.

#### Article information

Source
Adv. in Appl. Probab., Volume 48, Number 2 (2016), 585-609.

Dates
First available in Project Euclid: 9 June 2016

https://projecteuclid.org/euclid.aap/1465490764

Mathematical Reviews number (MathSciNet)
MR3511777

Zentralblatt MATH identifier
1346.60015

#### Citation

Acan, Hüseyin; Hitczenko, Paweł. On a memory game and preferential attachment graphs. Adv. in Appl. Probab. 48 (2016), no. 2, 585--609. https://projecteuclid.org/euclid.aap/1465490764

#### References

• Acan, H. and Hitczenko, P. (2015). On the covariances of outdegrees in random plane recursive trees. J. Appl. Prob. 52, 904–907.
• Acan, H. and Pittel, B. (2014). Formation of a giant component in the intersection graph of a random chord diagram. Preprint. Available at http://arxiv.org/abs/1406.2867.
• Andersen, J. E., Penner, R. C., Reidys, C. M. and Waterman, M. S. (2013). Topological classification and enumeration of RNA structures by genus. J. Math. Biol. 67, 1261–1278.
• Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.
• Berger, N., Borgs, C., Chayes, J. T. and Saberi, A. (2014). Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Prob. 42, 1–40.
• Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.
• Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290.
• Chmutov, S., Duzhin, S. and Mostovoy, J. (2012). Introduction to Vassiliev Knot Invariants. Cambridge University Press.
• Cori, R. and Marcus, M. (1998). Counting non-isomorphic chord diagrams. Theoret. Comput. Sci. 204, 55–73.
• Dulucq, S. and Penaud, J.-G. (1993). Cordes, arbres et permutations. Discrete Math. 117, 89–105.
• Flajolet, P. and Noy, M. (2000). Analytic combinatorics of chord diagrams. In Formal Power Series and Algebraic Combinatorics, Springer, Berlin, pp. 191–201.
• Graham, R. L., Knuth, D. E. and Patashnik, O. (1994). Concrete Mathematics, 2nd edn. Addison-Wesley, Reading, MA.
• Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177–245.
• Janson, S. (2005). Asymptotic degree distribution in random recursive trees. Random Structures Algorithms 26, 69–83.
• Peköz, E., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Prob. 23, 1188–1218.
• Peköz, E. E., Ross, N. and Röllin, A. (2014). Joint degree distributions of preferential attachment random graphs. Preprint. Available at http://arxiv.org/abs/1402.4686v1.
• Riordan, J. (1975). The distribution of crossings of chords joining pairs of $2n$ points on a circle. Math. Comp. 29, 215–222.
• Ross, N. (2013). Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Adv. Appl. Prob. 45, 876–893.
• Stoimenow, A. (1998). Enumeration of chord diagrams and an upper bound for Vassiliev invariants. J. Knot Theory Ramifications 7, 93–114.
• Velleman, D. J. and Warrington, G. S. (2013). What to expect in a game of memory. Amer. Math. Monthly 120, 787–805.
• Zhang, L.-X., Hu, F., Chung, S. H. and Chan, W. S. (2011). Immigrated urn models–-theoretical properties and applications. Ann. Statist. 39, 643–671.