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June 2016 On a memory game and preferential attachment graphs
Hüseyin Acan, Paweł Hitczenko
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Adv. in Appl. Probab. 48(2): 585-609 (June 2016).

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.

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Hüseyin Acan. Paweł Hitczenko. "On a memory game and preferential attachment graphs." Adv. in Appl. Probab. 48 (2) 585 - 609, June 2016.

Information

Published: June 2016
First available in Project Euclid: 9 June 2016

zbMATH: 1346.60015
MathSciNet: MR3511777

Subjects:
Primary: 60F05
Secondary: 05C82 , 60B12 , 60C05

Keywords: Convergence in distribution , generalized Pólya urn , memory game , preferential attachment graph

Rights: Copyright © 2016 Applied Probability Trust

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Vol.48 • No. 2 • June 2016
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