P\'olya Urn Schemes with Infinitely Many Colors

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 non-zero centering is always ${\mathcal O}\left(\log n\right)$ and the scaling is ${\mathcal O}\left(\sqrt{\log n}\right)$. The work also provides similar results for urn models with infinitely many colors indexed by more general lattices in ${\mathbb R}^d$. 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.Comment: 32 pages and 1 figure. Motivation for the work has been added and few typing errors have been correcte

Nonuniform random geometric graphs with location-dependent radii

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

Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

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. 19 (2009) 1972-2007] 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. 35 (2007) 1783-1806] and we settle a conjecture of Sellke (1994) by showing that for any reciprocally summable reinforcement weight function w, the walk traverses a random attracting edge at all large times. For vertex-reinforced random walks, we extend results previously obtained on Z by Volkov [Ann. Probab. 29 (2001) 66-91] and by Basdevant, Schapira and Singh [Ann. Probab. 42 (2014) 527-558], and on complete graphs by Benaim, Raimond and Schapira [ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 767-782]. 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

Positive reinforced generalized time-dependent Pólya urns via stochastic approximation

Consider a generalized time-dependent Pólya urn process defined as follows. Let d∈N be the number of urns/colors. At each time n, we distribute σn balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions R assuming some monotonicity and growth condition. The class R includes convex functions and the classical case f(x)=xα, α>1. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls any more

A new approach to Pólya urn schemes and its infinite color generalization

In this work, we introduce a generalization of the classical Pólya urn scheme (Ann. Inst. Henri Poincaré 1 (1930) 117–161) with colors indexed by a Polish space, say, S. The urns are defined as finite measures on S endowed with the Borel σ-algebra, say, S . The generalization is an extension of a model introduced earlier by Blackwell and MacQueen (Ann. Statist. 1 (1973) 353–355). We present a novel approach of representing the observed sequence of colors from such a scheme in terms an associated branching Markov chain on the random recursive tree. The work presents fairly general asymptotic results for this new generalized urn models. As special cases, we show that the results on classical urns, as well as, some of the results proved recently for infinite color urn models in (Bernoulli 23 (2017) 3243–3267; Statist. Probab. Lett. 92 (2014) 232–240), can easily be derived using the general asymptotic. We also demonstrate some newer results for infinite color urns

Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

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. 19 (2009) 1972-2007] 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. 35 (2007) 1783-1806] and we settle a conjecture of Sellke (1994) by showing that for any reciprocally summable reinforcement weight function w, the walk traverses a random attracting edge at all large times. For vertex-reinforced random walks, we extend results previously obtained on Z by Volkov [Ann. Probab. 29 (2001) 66-91] and by Basdevant, Schapira and Singh [Ann. Probab. 42 (2014) 527-558], and on complete graphs by Benaim, Raimond and Schapira [ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 767-782]. 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