On a memory game and preferential attachment graphs

Abstract

In a recent paper Velleman and Warrington 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 simpler proofs (although without rate of convergence) of some of the results of Peköz, Röllin, and Ross on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs.For proving 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 multi-type generalized Pólya urn model.

Probabilistic consequences of some polynomial recurrences

“…We adapt a result of [17] to do this in our particular situation which may be useful in other settings. In addition, this paper discusses similar types of polynomial recurrences which are found in the literature (see [1,6,12,15]). This paper establishes a framework to approach such recurrences and derive the probabilistic consequences of them.…”

“…In addition, this paper discusses similar types of polynomial recurrences which are found in the literature (see). This paper establishes a framework to approach such recurrences and derive the probabilistic consequences of them.…”

“…This is a generalization of the classical Eulerian polynomials. Specifically, the choice of parameters a = 1 and b = 0 gives P n, 1,0…”

“…As one more example, the following recurrence was used in [1,Section 3] in connection with the analysis of a version of a card game called the memory game.…”

“…The limiting distribution of a sequence (X n ) of random variables associated with the recurrence (13) was found in [1, Section 3] by elementary counting and did not rely on (13). In fact, the array of coefficients of the polynomials A n (x) appeared in the literature significantly earlier than [1] or [21] (see, eg, [5]) and is featured in [20] as sequence A102625. Specifically, in the notation of [20], T(n, k), 1 ≤ k ≤ n + 1 is the coefficient of x k+1 in A n+1 (x) (and it corresponds to a(n + 1, k + 1) in the notation of [21]).…”