Abstract. In this work we introduce a new type of urn model with infinite but countable many colors indexed by an appropriate infinite set. We mainly consider the indexing set of colors to be the d-dimensional integer lattice and consider balanced replacement schemes associated with bounded increment random walks on it. We prove central and local limit theorems for the random color of the n-th selected ball and show that irrespective of the null recurrent or transient behavior of the underlying random walks, the asymptotic distribution is Gaussian after appropriate centering and scaling. We show that the order of any nonzero centering is always O (log n) and the scaling is O √ log n . The work also provides similar results for urn models with infinitely many colors indexed by more general lattices in R d . We introduce a novel technique of representing the random color of the n-th selected ball as a suitably sampled point on the path of the underlying random walk. This helps us to derive the central and local limit theorems.

a b s t r a c tIn this work we consider the infinite color urn model associated with a bounded increment random walk on Z d . This model was first introduced in Bandyopadhyay and Thacker (2013). We prove that the rate of convergence of the expected configuration of the urn at time n with appropriate centering and scaling is of the order O  (log n) −1/2  . Moreover we derive bounds similar to the classical Berry-Esseen bound. Further we show that for the expected configuration a large deviation principle (LDP) holds with a good rate function and speed log n.

In this paper we introduce a new simple but powerful general technique for the study of edge-and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. The technique relies on upper bound estimates for the number of edge traversals, proved in a different context by Cotar and Limic [Ann. Appl. Probab. (2009)] for finite graphs with edge reinforcement. We apply our new method both to edge-and to vertex-reinforced random walks with super-linear reinforcement on arbitrary infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with super-linear reinforcement, we make no other assumption on the graphs.For edge-reinforced random walks, we complete the results of Limic and Tarrès [Ann. Probab. (2013)]. We show that on any infinite connected graph of bounded degree, with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions, the walk traverses two random neighbouring attracting vertices at all large times.

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We consider the generalization of the Pólya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under the uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.

Start with a graph with a subset of vertices called the border. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset thus increasing the border by one point. Then a new particle is released from the origin and the process repeats until the origin becomes a part of the border itself. We are interested in the total number ξ of particles to be released by this final moment.We show that this model covers OK Corral model as well as the erosion model, and obtain distributions and bounds for ξ in cases where the graph is star graph, regular tree, and a d−dimensional lattice.

We propose a distribution-free approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function $nf(\cdot)$, where $n\in \mathbb{N}$, and $f$ is a probability density function on $\mathbb{R}^d$. A vertex located at $x$ connects via directed edges to other vertices that are within a cut-off distance $r_n(x)$. We prove strong law results for (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large $n$ and (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.Comment: Published in at http://dx.doi.org/10.1214/11-AAP823 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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