In this paper, we consider a multi-drawing urn model with random addition. At each discrete time step, we draw a sample of m balls. According to the composition of the drawn colors, we return the balls together with a random number of balls depending on two discrete random variables X and Y with finite means and variances. Via the stochastic approximation algorithm, we give limit theorems describing the asymptotic behavior of white balls.
We consider a version of the classical Pólya 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 additional balls of different colors given by some finite point process $ξ$ on $S$. When the number of steps goes to infinity, the empirical distribution of the colors in the urn converges to the normalized intensity measure of $ξ$, and we analyze the fluctuations. The ratio $ρ= E(C)/E(R)$ of the average number of copies to the average total number of balls returned p…
We consider a time-dependent version of a Pólya urn containing black and white balls. At each time $n$ a ball is drawn from the urn at random and replaced in the urn along with $σ_n$ additional balls of the same colour. The proportion of white balls converges almost surely to a random limit $Θ$, and $\mathcal{D}=\{Θ\in\{0,1\}\}$ denotes the event when one of the colours dominates. The phase transition, in terms of the sequence $(σ_n)$, between the regimes ${\mathbb P}(\mathcal{D})=1$ and ${\mathbb P}(\mathcal{D})<1$ was obtained by R. Pemantle in 1990. We describe the phase transition between …
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 proportional to $w_k$ where $k$ is the number of balls of this colour. A classical result states that an urn fixates on one colour after a finite time if an only if $\sum_{0}^\infty w_k^{-1} < \infty$. In this paper we shall study the case when at each time unit we draw with replacement a number $d\in\N$ of balls and then add $d$ new balls of matching colours. The main…
We study the number of white balls in a classical Pólya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times are i.i.d. and, under certain moment conditions on the inter-arrival distribution, we characterize the limiting distribution of the (properly scaled) number of white balls as the number of draws goes to infinity. The possible limiting distributions obtained in this way vary considerably depending on the inter-arrival distribution and are difficult to describe explicitly. However, we show that the li…
In the classical Polya urn problem, one begins with $d$ bins, each containing one ball. Additional balls arrive one at a time, and the probability that an arriving ball is placed in a given bin is proportional to $m^γ$, where $m$ is the number of balls in that bin. In this note, we consider the case of $γ= 1$, which corresponds to a process of "proportional preferential attachment" and is a critical point with respect to the limit distribution of the fraction of balls in each bin. It is well known that for $γ< 1$ the fraction of balls in the "heaviest" bin (the bin with the most balls) tends t…
This is a research endeavor in two parts. We study a class of balanced urn schemes on balls of two colours (say white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn, and ball addition rules are applied. We consider these multiple drawings under sampling with or without replacement. We further classify ball addition matrices according to the structure of the expected value into affine and nonaffine classes. We give a necessary and sufficient condition for a scheme to be in the affine subclass. For the affine subclass, we get explicit results for the expected value …
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 general random entries. For the proportion of balls of a given color, we prove almost sure convergence results and fluctuation theorems (through CLTs in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution). Asymptotic confidence intervals are given for the limit proportion, whose distribut…
For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the "power law Pólya's urn" of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three a…
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 are applied; the special case $m=1$ of sampling only a single ball coincides with ordinary balanced urn models. We consider these multiple drawings under sampling with or without replacement. For the class of affine conditional expected value, we study the number of white balls after $n$ steps. The affine class is parametrized by $Λ$, specified by the ratio…
The Pólya urn scheme is a discrete-time process concerning the addition and removal of colored balls. There is a known embedding of it in continuous-time, called the Pólya process. We deal with a generalization of this stochastic model, where the initial values and the entries of the transition matrix (corresponding to additions or removals) are not necessarily fixed integer values as in the standard Pólya process. In one of the scenarios, we even allow the entries of the matrix to be random variables. As a result, we no longer have a combinatorial model of "balls in an urn," but a broader int…
We study first passage statistics of the Polya urn model. In this random process, the urn contains two types of balls. In each step, one ball is drawn randomly from the urn, and subsequently placed back into the urn together with an additional ball of the same type. We derive the probability G_n that the two types of balls are equal in number, for the first time, when there is a total of 2n balls. This first passage probability decays algebraically, G_n ~ n^{-2}, when n is large. We also derive the probability that a tie ever happens. This probability is between zero and one, so that a tie may…
Pólya urns are urns where at each unit of time a ball is drawn and 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 of the time ($\operatorname{mod} p$). We extend the work of Flajolet et al. on Pólya urns: the generating function encoding the evolution of the urn is studied by methods of analytic combinatorics. We show that the initial partial differential equations lead to ordinary linear differential equations which are related to hypergeometric functions (giving the exact st…
In the present paper we show that in Pólya's urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the absolute value of the error of Bernstein-Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proof relies on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first mome…
Since its inception in 1907, the Ehrenfest urn model (EUM) has served as a test bed of key concepts of statistical mechanics. Here we employ this model to study large deviations of a time-additive quantity. We consider two continuous-time versions of the EUM with $K$ urns and $N$ balls: without and with interactions between the balls in the same urn. We evaluate the probability distribution $\mathcal{P}_T(\bar{n}= a N)$ that the average number of balls in one urn over time $T$, $\bar{n}$, takes any specified value $aN$, where $0\leq a\leq 1$. For long observation time, $T\to \infty$, a Donsker…
An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L<U\leq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\overset{a.s.}\longrightarrow Z$ for some random variab…
