G. Aletti, C. May, P. Secchi
We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.
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We give a central limit theorem, which has applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced generalized Polya urns.
We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color given the past is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is…
This paper illustrates asymptotic properties for a response-adaptive design generated by a two-color, randomly reinforced urn model. The design considered is optimal in the sense that it assigns patients to the best treatment, with probability converging to one. An approach to show the joint asymptotic normality of the estimators of the mean responses to the treatments is provided in spite of the fact that allocation proportions converge to zero and one. Results on the rate of convergence of the number of patients assigned to each treatment are also obtained. Finally, we study the asymptotic b…
We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic approximation. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field r…
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We prove a Gaussian process approximation for the sequence of random compositions of a two-color randomly reinforced urn for both the cases with the equal and unequal reinforcement means. By using the Gaussian approximation, the law of the iterated logarithm and the functional limit central limit theorem in both the stable convergence sense and the almost-sure conditional convergence sense are established. Also as a consequence, we are able to to prove that the distribution of the urn composition has no points masses both when the reinforcement means are equal and unequal under the assumption …
We consider weighted negatively reinforced urn schemes with finitely many colours. An urn scheme is called negatively reinforced, if the selection probability for a colour is proportional to the weight $w$ of the colour proportion, where $w$ is a non-increasing function. Under certain assumptions on the replacement matrix $R$ and weight function $w$, such as, $w$ is differentiable and $w(0) < \infty$, we obtain almost sure convergence of the random configuration of the urn model. In particular, we show that if $R$ is doubly stochastic the random configuration of the urn converges to the unifor…
Though widely used in applications, reinforced random walk on graphs have never been the subject of a valid statistical inference. We develop in this paper a statistical framework for a general two-colored urn model. The probability to draw a ball at each step depends on the number of balls of each color and on a multidimensional parameter $θ$ through a function $f$, called a choice function. We introduce two estimators of $θ$: the maximum likelihood estimator and a weighted least squares estimator which is less efficient, but is closer to the calibration techniques used in the applied literat…
This work deals with a system of interacting reinforced stochastic processes, where each process $X^j=(X_{n,j})_n$ is located at a vertex $j$ of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent $j$ of the network. The interaction among the dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal "inclination" $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h\neq j$, of the other agents accordi…
Consider a generalized time-dependent Pólya urn process defined as follows. Let $d\in \mathbb{N}$ be the number of urns/colors. At each time $n$, we distribute $σ_n$ 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 $\mathcal{R}$ assuming some monotonicity and growth condition. The class $\mathcal{R}$ includes convex functions and the classical case $f(x)=x^α$, $α>1$. The novelty of the paper lies in extending stochastic approximation techniques to the $d$-dimensional case and prov…
The generalized Pòlya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with diagonal replacement matrix, which arise in several applications, specifically in clinical trials. To facilitate mathematical analyses of models in these applications, we introduce an adaptive randomly reinforced urn model that uses accruing statistical information to adaptively skew the urn proportion toward specific targets. We study several probabilistic aspects that are important in implementing the…
Adaptive randomly reinforced urn (ARRU) is a two-color urn model where the updating process is defined by a sequence of non-negative random vectors $\{(D_{1,n}, D_{2,n});n\geq1\}$ and randomly evolving thresholds which utilize accruing statistical information for the updates. Let $m_1=E[D_{1,n}]$ and $m_2=E[D_{2,n}]$. Motivated by applications, in this paper we undertake a detailed study of the dynamics of the ARRU model. First, for the case $m_1 \neq m_2$, we establish $L_1$ bounds on the increments of the urn proportion at fixed and increasing times under very weak assumptions on the random …
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…
We consider, as proposed and studied in Hofstad et.\ al.\ \cite{HHKR}, a class of graph-based "interacting urn"-type Polya urn model inspired by neuronal processing in the brain where a signal enters the brain at some (randomly) chosen neuron and is transmitted to a (random) single neighbouring neuron with a probability depending on the relative `efficiency' of the synapses connecting the neurons, and in doing so the efficiency of the utilized synapse is improved/reinforced. We study the structures (or architectures) and relative efficiency of the neuronal networks that can arise from repeatin…
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 of, the number of times that particle traversed that edge, a quantity which depends on the behaviour of the other particles, and possibly external factors. We study localization properties of this system and our main tool is a new result we establish for a very general class of urn models. More specifically, we study attraction properties of urns composed o…
Vertex reinforced branching random walks and generalized time-dependent Polya urns
研究整数集上受顶点强化的无限临界树索引随机游走，关注强强化情形。通过分析时间依赖广义波利亚瓮过程，对每步抽取次数由序列规定、选球概率与球数函数相关的情况进行研究，有界序列时恢复鲁宾颜色固定特征描述。
Infinite-color randomly reinforced urns with dominant colors
本文定义具有无界颜色数和颜色特定权重的主导 Pólya 随机强化罐，证明预测与经验分布收敛到集中于最大期望强化颜色集的随机概率测度，并给出速率及应用。
The models surveyed include generalized Pólya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.
The aim of this paper is to study the asymptotic behavior of strongly reinforced interacting urns with partial memory sharing. The reinforcement mechanism considered is as follows: draw at each step and for each urn a white or black ball from either all the urns combined (with probability $p$) or the urn alone (with probability $1-p$) and add a new ball of the same color to this urn. The probability of drawing a ball of a certain color is proportional to $w_k$ where $k$ is the number of balls of this color. The higher the $p$, the more memory is shared between the urns. The main results can be…
In this paper, we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existing colours. We refer to it as the \emph{negatively reinforced} urn scheme. We establish almost sure limit of the random configuration for any \emph{balanced} replacement matrix $R$. In particular, we show that the limiting configuration is uniform on the set of colours, if and only if, $R$ is a \emph{doubly stochastic} matrix. We further establish almost sure limit of the vector of colour counts and prove central…
On Pólya-Young urn models and growth processes
本文推广了Pólya-Young罐模型，利用噪声增强Bessel过程局部时给出极限律，通过鞅结构得到中心极限定理和重对数律，并建立其与递增树、Stirling排列推广及带竞争中国餐馆过程的联系。
A central limit theorem for unbalanced step-reinforced random walks
本文研究一类统一了象随机游走等模型的不平衡步长强化随机游走，通过建立与随机递归树键渗流的联系将其表示为随机加权和，推导出正态与稳定中心极限定理，进而获得该类游走的统一中心极限定理。
Variations on Reinforced Random Walks
本论文研究边强化随机游走的变体，包括多游走者相互作用和方向偏置，证明在特定条件下出现常返与暂态间的相变，并揭示与标准模型的根本差异。
This paper deals with different models of random walks with a reinforced memory of preferential attachment type. We consider extensions of the Elephant Random Walk introduced by Schütz and Trimper [2004] with a stronger reinforcement mechanism, where, roughly speaking, a step from the past is remembered proportional to some weight and then repeated with probability $p$. With probability $1-p$, the random walk performs a step independent of the past. The weight of the remembered step is increased by an additive factor $b\geq 0$, making it likelier to repeat the step again in the future. A combi…
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…
An urn model for opinion propagation on networks
本文提出网络社会动力学中的耦合 Polya 坛子模型，证明其由随机热方程的离散版本控制，并最终达成意见共识。
Interacting Edge-Reinforced Random Walks
研究多个游走者共同影响边权重的边强化随机游走，证明其行为与单个游走者一致，并给出常返性判据。
Berry-Esseen bounds for step-reinforced random walks
本文研究带参数$p$的正、负向阶跃强化随机游走，在步长分布三阶矩有限的假设下，通过建立独立随机变量泛函的Berry-Esseen界等方法，得到了两类游走收敛于正态分布的速率。
First to reach $n$ game
研究一个包含两玩家的多轮游戏，先赢 $n$ 轮者胜，每轮胜负由从特定瓮中抽取的球类型决定，分析三种抽取规则下玩家净收益的随机变量性质。
On a class of unbalanced step-reinforced random walks
提出一类非平衡步进增强随机游走，证明其强大数定律和中心极限定理，统一了大象随机游走及正/负步进增强随机游走的处理。
Conditional identity in distribution (Berti et al. (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper, a class of random sequences, called Generalized Species Sampling Sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, a class of generalized species sampling sequences that are conditionally identically distributed is introduced and studied: the Generalized Ottawa sequences (GOS). This class contains a '`randomly reinforced'' version of the Pólya urn and of the Bl…
We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight $\vartheta>0$, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For $\vartheta=1$ the resulting tree is the well-studied random recursive tree. We analyze the height,…
We consider two types of random networks grown in blocks. Hooking networks are grown from a set of graphs as blocks, each with a labelled vertex called a hook. At each step in the growth of the network, a vertex called a latch is chosen from the hooking network and a copy of one of the blocks is attached by fusing its hook with the latch. Bipolar networks are grown from a set of directed graphs as blocks, each with a single source and a single sink. At each step in the growth of the network, an arc is chosen and is replaced with a copy of one of the blocks. Using Pólya urns, we prove normal li…
Randomly evolving systems composed by elements which interact among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon, that could be roughly defined as the tendency of different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement, that recently has been introduced in Crimaldi et al. (2016, arXiv:1602.06217). Generally speaking, by reinforcement we mean any mechanism for which the probability that a given event occurs has an increasing dependen…
Exact random centering and hybrid fluctuation limits for interacting reinforced processes under critical Markov switching
针对临界马尔可夫切换下交互强化过程，提出精确随机中心化与坐标分解，证明混合PDMP-扩散极限并刻画平稳性质，恢复交互信息。
We study the limit behaviour of a class of random walk models taking values in the $d$-dimensional unit standard simplex, $d\ge 1$, defined as follows. From an interior point $z$, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on $z$, and then the particle randomly jumps to a new location $z'$ on the segment connecting $z$ to the chosen vertex. In some specific cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are, in fact, Dirichlet. We also consider a related history-dependent random walk…
Localisation of the monkey walk with an "impurity" at the origin
本文研究带有原点处“杂质”的猴子行走模型，证明当非原点处重定位概率足够接近1时，该模型会发生定位，而非原点处重定位概率等于原点处时则会扩散。
Some developments of exchangeable measure-valued Pólya sequences
本文研究可交换测度值Pólya序列，证明其先验分布是Dirichlet过程混合，并扩展到包含零分量和条件同分布情形。
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