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We consider a version of the classical P\'olya urn scheme which incorporates innovations. The space $S$ of colors is an arbitrary measurable set. After each sampling of a ball in the urn, one returns $C$ balls of the same color and…

The asymptotic behaviour of a generalised P\'olya--Eggenberger urn is well--known to depend on the spectrum of its replacement matrix: If its dominant eigenvalue $r$ is simple and no other eigenvalue is `large' in the sense that its real…

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn…

This paper considers a two-color, single-draw urn model with two types of balls, denoted type $1$ and type $2$, with initial counts $Y^1_0\in N^+$ and $Y^2_0\in N^+$, respectively. At each discrete time step, a ball is drawn uniformly at…

Let $G$ be a finite Abelian group of order $d$. We consider an urn in which, initially, there are labeled balls that generate the group $G$. Choosing two balls from the urn with replacement, observe their labels, and perform a group…

We study a generalized P\'{o}lya urn model with two types of ball. If the drawn ball is red, it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black…

P{\'o}lya urns are urns where at each unit of time a ball is drawn and is replaced with some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value…

We introduce a modification of the generalized P\'olya urn model containing two urns and we study the number of balls $B_j(n)$ of a given color $j\in\{1,\ldots,J\}$, $J\in\mathbb{N}$ added to the urns after $n$ draws. We provide sufficient…

This is the second part of a two-part investigation. We continue the study of a class of balanced urn schemes on balls of two colors (white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn and ball addition rules…

We consider a mean-field dynamical urn model, defined by rules which give the rate at which a ball is drawn from an urn and put in another one, chosen amongst an assembly. At equilibrium, this model possesses a fluid and a condensed phase,…

We consider multicolor urn models with multiple drawings. An urn model is called linear if the conditional expected value of the urn composition at time $n$ is a linear function of the composition at time $n-1$. For four different sampling…

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…

In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: $(\theta,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \begin{eqnarray} \nonumber f_{\pm 1}:…

In classical urn models, one usually draws one ball with replacement at each time unit and then adds one ball of the same colour. Given a weight sequence $(w_k)_{k\in\N}$, the probability of drawing a ball of a certain colour is…

This article is dedicated to the study of the normal functor in the category of smooth real vector bundles. Particularly, we focus on a symmetry phenomena which occurs after iterating two times the normal functor on a commutative square of…

We extend a general result showing that the asymptotic behavior of high moments, factorial or standard, of random variables, determines the asymptotically normality, from the one dimensional to the multidimensional setting. This approach…

The symmetric random walk is known to be recurrent in one and two dimensions, and becomes transient in three or higher dimensions. We compare the symmetric random walk to walks driven by certain \polya\ urns. We show that, in contrast, if…

Strongly consistent estimates are shown, via relative frequency, for the probability of "white balls" inside a dichotomous urn when such a probability is an arbitrary continuous time dependent function over a bounded time interval. The…

Consider a number, finite or not, of urns each with fixed capacity $r$ and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain $r$ balls. When $r=1$, using analytic…

We consider a two-color P\'{o}lya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a large urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings,…