Related papers: P\'olya urn with memory kernel and asymptotic beha…
P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference…
We study the phase transition and the critical properties of a nonlinear P\'{o}lya urn, which is a simple binary stochastic process $X(t)\in \{0,1\},t=1,\cdots$ with a feedback mechanism. Let $f$ be a continuous function from the unit…
In the present paper we prove that the probabilities of the P\'olya urn distribution (with negative replacement) satisfy a monotonicity property similar to that of the binomial distribution (P\'olya urn distribution with no replacement). As…
We consider an urn model with multiple drawing and random time-dependent addition matrix. The model is very general with respect to previous literature: the number of sampled balls at each time-step is random, the addition matrix has…
Generating realistic artificial preference distributions is an important part of any simulation analysis of electoral systems. While this has been discussed in some detail in the context of a single electoral district, many electoral…
We consider a system of urns of Polya-type, with balls of two colors; the reinforcement of each urn depends both on the content of the same urn and on the average content of all urns. We show that the urns synchronize almost surely, in the…
We consider a special case of the generalized P\'{o}lya's urn model introduced by Benaim et al (2013). Given a finite connected graph $G$, place a bin at each vertex. Two bins are called a pair if they share an edge of $G$. At discrete…
Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability…
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…
In this work, recent results on the moments of balanced P\'olya urns are generalized to unbalanced urns, with the condition that the expected change in total activity at each step is constant. We also provide applications of our results to…
In this paper, we prove convergence and fluctuation results for measure-valued P\'olya processes (MVPPs, also known as P\'olya urns with infinitely-many colours). Our convergence results hold almost surely and in $L^2$, under assumptions…
We show that the supremum of the successive percentages of red balls in P\'olya's urn model is almost surely rational, give the set of values that are taken with positive probability and derive several exact distributional results for the…
Balls and bins models are classical probabilistic models where balls are added to bins at random according to a certain rule. The balls and bins model with feedback is a non-linear generalisation of the P\'olya urn, where the probability of…
We propose an elementary but effective approach to studying a general class of Poissonized tenable and balanced urns on two colors. We characterize the asymptotic behavior of the process via a partial differential equation that governs the…
Following Hales (2018), the evolution of P\'olya's urn may be interpreted as a walk, a P\'olya walk, on the integer lattice $\mathbb{N}^2$. We study the visibility properties of P\'olya's walk or, equivalently, the divisibility properties…
We study a system of interacting reinforced random walks defined on polygons. At each stage, each particle chooses an edge to traverse which is incident to its position. We allow the probability of choosing a given edge to depend on the sum…
It is known that in an irreducible small P\'olya urn process, the composition of the urn after suitable normalization converges in distribution to a normal distribution. We show that if the urn also is balanced, this normal convergence…
For the most general Polya urn schemes, we establish the almost sure convergence of its composition. The only requirement is that there are always enough balls of both colors, so that the extractions can be indefinitely pursued according to…
We study the long-time asymptotics of a network of weakly reinforced P\'olya urns. In this system, which extends the WARM introduced by R. van der Hofstad et. al. (2016) to countable networks, the nodes fire at times given by a Poisson…
Generalized P\'olya urns with non-linear feedback are an established probabilistic model to describe the dynamics of growth processes with reinforcement, a generic example being competition of agents in evolving markets. It is well known…