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On a memory game and preferential attachment graphs

Part of: Graph theory Probability theory on algebraic and topological structures Combinatorial probability Limit theorems

Published online by Cambridge University Press:  10 June 2016

Hüseyin Acan  and
Paweł Hitczenko
Hüseyin Acan*
Affiliation:
Monash University
Paweł Hitczenko*
Affiliation:
Drexel University
*
* Current address: Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA. Email address: huseyin.acan@rutgers.edu
** Postal address: Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA. Email address: phitczen@math.drexel.edu
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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.

Keywords

Convergence in distributiongeneralized Pólya urnpreferential attachment graphmemory game

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 05C82: Small world graphs, complex networks 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case) 60C05: Combinatorial probability
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 585 - 609
Copyright
Copyright © Applied Probability Trust 2016 

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