Limit Theorems for Random Triangular URN Schemes

Abstract

In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.

Elephant random walks and their connection to Pólya-type urns

“…with e 1 = (1, 0) and e 2 = (0, 1). Starting from X 1 = (1, 0, 0, 0), say, this means that the ERW first visits (1,0). Then, at any later time n ≥ 2, the walker chooses a time n ′ uniformly at random among the previous times 1, .…”

“…The eigenvalues of the above matrix C are λ 1 = 1 and λ 2 = p. Here, the results of [15] are not applicable, since λ 1 does not belong to the dominating class: Indeed, when starting the urn process from a single red ball, the dynamics adds only red balls to the urn, and never a black ball. Such random triangular urn schemes were however treated by Aguech [1], generalizing results of Janson [16] for triangular urns with deterministic replacement. In particular, [1, Theorem 2(a)] shows that the right rescaling for the number of black balls at time n is n p (there is no recentering), and one has almost sure-convergence as n tends to infinity to a nontrivial (non-Gaussian) limit.…”

“…A simple model exhibiting anomalous diffusion is the so-called Elephant Random Walk (ERW) introduced by Schütz and Trimper [28] in 2004, which is the topic of this paper. The ERW model is a one-dimensional discrete-time nearest neighbor random walk on Z, which remembers its full history and chooses its next step as follows: First, it selects randomly a step from the past, and then, with probability p ∈ [0, 1], it repeats what it did at the remembered time, whereas with the complementary probability 1 − p, it makes a step in the opposite direction. We refer to the next section for the precise definition.…”

Unbalanced multi-drawing urn with random addition matrix

“…The case when µ X < µ Y is obtained by interchanging the colors. Example: Let m = 1, this particular case was studied by R. Aguech [16]. Using martingales and branching processes , R. Aguech proved the following results:…”

“…In the present paper, we deal with an unbalanced urn model, which was not been sufficiently addressed in the literature. It was mainly dealt with in the works of R. Aguech [16], S. Janson [19] and H. Renlund [4,5]. In [16] and [19], the authors dealt with model with a simple pick, whereas in [4,5] the author considered a model with two picks and, under some conditions, they described the asymptotic behavior of the urn composition.…”