Advances in Applied Probability

On a memory game and preferential attachment graphs

Hüseyin Acan and Paweł Hitczenko

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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

Permanent link to this document
https://projecteuclid.org/euclid.aap/1465490764

Mathematical Reviews number (MathSciNet)
MR3511777

Zentralblatt MATH identifier
1346.60015

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30] 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60C05: Combinatorial probability

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

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


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