Mikhail Chebunin, Artyom Kovalevskii
We investigate a specific infinite urn scheme first considered by Karlin (1967). We prove functional central limit theorems for the total number of urns with at least k balls for different k.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labelled 1,2,... so that the urn $j$ at every draw gets a ball with probability $p_j$, $\sum_j p_j=1$. We prove functional central limit theorems for discrete time and the poissonised version for the urn occupancies process, for the odd-occupancy and for the missing mass processes extending the known non-functional central limit theorems.
In this work we generalize Polya urn schemes with possibly infinitely many colors and extend the earlier models described in [4, 5, 7]. We provide a novel and unique approach of representing the observed sequence of colors in terms a branching Markov chain on random recursion tree. This enables us to derive fairly general asymptotic for our urn schemes. We then illustrate through several examples that our method can easily derive the classical results for finite urns, as well as, many new results for infinite color urns.
Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation.
We consider the generalization of the Pólya urn scheme with possibly infinite many colors as introduced in \cite{Th-Thesis, BaTH2014, BaTh2016, BaTh2017}. For countable many colors, we prove almost sure convergence of the urn configuration under \emph{uniform ergodicity} assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a \emph{branching Markov chain} on a weighted \emph{random recursive tree} as described in \cite{BaTh2017, Sv_2018}. Using this coupling we estimate the covariance between any two selected colors. In particular…
In this work we consider the \emph{infinite color urn model} associated with a bounded increment random walk on $\Zbold^d$. This model was first introduced by Bandyopadhyay and Thacker (2013). We prove that the rate of convergence of the expected configuration of the urn at time $n$ with appropriate centering and scaling is of the order ${\mathcal O}\left(\frac{1}{\sqrt{\log n}}\right)$. Moreover we derive bounds similar to the classical Berry-Essen bound. Further we show that for the expected configuration a \emph{large deviation principle (LDP)} holds with a good rate function and speed $\lo…
We investigate the randomized Karlin model with parameter $β\in(0,1)$, which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index $β/2\in(0,1/2)$. We show here that when the randomization is heavy-tailed with index $α\in(0,2)$, then the odd-occupancy process scales to a $(β/α)$-self-similar symmetric $α$-stable process with stationary increments.
Limit theorems for a class of random outer measures in infinite urn schemes
本文研究了无限尿方案中一类随机外测度的极限定理，通过函数中心极限定理分析了文本概率模型的应用。
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 tha…
We study the joint asymptotics of forward and backward processes of numbers of non-empty urns in an infinite urn scheme. The probabilities of balls hitting the urns are assumed to satisfy the conditions of regular decrease. We prove weak convergence to a two-dimensional Gaussian process. Its covariance function depends only on exponent of regular decrease of probabilities. We obtain parameter estimates that have a normal asymototics for its joint distribution together with forward and backward processes. We use these estimates to construct statistical tests for the homogeneity of the urn schem…
We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to the decomposition of a time-changed Brownian motion $\mathbb B(t^α), α\in(0,1)$, and the rand…
The random vector of frequencies in a generalized urn model is viewed as conditionally independent random variables, given their sum. Such a representation is exploited to derive Edgeworth expansions for a sum of functions of such frequencies. Applying these results to urn models such as with- and without-replacement sampling schemes as well as the multicolor Polya-Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without replacement scheme, and for the so-called Dixon statistic that is useful in comparing two samples.
We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n>0, choose a ball from the urn uniformly at random. With probability 1/2<p<1, return the ball to the urn along with another ball of the same colour. With probability 1-p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri, and Thörnblad. We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there i…
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。
Let X be a non-deterministic infinite exchangeable sequence with values in {0,1}. We show that X is Hoeffding-decomposable if, and only if, X is either an i.i.d. sequence or a Polya sequence. This completes the results established in Peccati [2004]. The proof uses several combinatorial implications of the correspondence between Hoeffding decomposability and weak independence. Our results must be compared with previous characterizations of i.i.d. and Polya sequences given by Hill et al. [1987] and Diaconis and Yilvisaker [1979].
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…
The Generalized Pólya Urn (GPU) is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we propose a sequential estimation-adjusted urn model (a nonhomogeneous GPU) which has a wide spectrum of applications. Because the proposed urn model depends on sequential estimations of unknown parameters, the derivation of asymptotic properties is mathematically intricate and the corresponding results are unavailable in the literature. We overcome these hurdles and establish the strong consis…
Triggered urn models for frequently asked questions (FAQ)
本文提出了一种带有触发机制的尿布模型，用于分析FAQ的分布特性，通过渐近分析和模拟验证了该模型在预测和实际应用中的有效性。
Quantitative limit theorems for generalized Pólya urns with applications to random tree models
研究广义波利亚瓮颜色渐近分布的定量极限定理，构建显式收敛速率。应用于随机递归树等三种模型，将出度统计转化为广义波利亚瓮，得出\(L^2\)和几乎必然意义下\(O(1/n)\)阶且出度一致的收敛速率。
Interacting Urn Schemes on Finite Ancestral Directed Acyclic Graphs
本文研究有限祖先有向无环图上的交互有限色瓮方案，证明了瓮比例的几乎必然收敛性，确定了极限构型的决定顶点，并在强化矩阵满足特定假设下得到二阶渐近结果。
An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some statistical applications are discussed.
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 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…
Tree shape statistics provide valuable quantitative insights into evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph representation of evolution systems ranging from viruses to species. By developing limit theorems for a version of extended Pólya urn models in which negative entries are permitted for their replacement matrices, we present strong laws of large numbers and central limit theorems for asymptotic joint distributions of two subtree counting statistics, the number of cherries and that of pitchforks, for random phylogenetic trees generated by two widely use…
The tools of zero biasing are adapted to yield a general result suitable for analyzing the behavior of certain growth processes. The main theorem is applied to prove central limit theorems, with explicit error terms in the L^1 metric, for certain statistics of the Jack measure on partitions and for the number of balls drawn in a Polya-Eggenberger urn process.
Consider a (possibly infinite) exchangeable sequence X={X_n:1\leqn<N}, where N\in N\cup {\infty}, with values in a Borel space (A,A), and note X_n=(X_1,...,X_n). We say that X is Hoeffding decomposable if, for each n, every square integrable, centered and symmetric statistic based on X_n can be written as an orthogonal sum of n U-statistics with degenerated and symmetric kernels of increasing order. The only two examples of Hoeffding decomposable sequences studied in the literature are i.i.d. random variables and extractions without replacement from a finite population. In the first part of th…
On central limit theorems for Ewens-Pitman model
本文基于中国餐馆过程研究了Ewens-Pitman模型中组件数量的中心极限定理，揭示了波动由采样效应和随机渐近频率波动两部分组成，且在极限条件下两者在给定α-多样性下条件独立。
Infinite ergodic theory and functional statistics of non-confined Feller process
本文研究非约束Feller过程局部时场的空间结构，证明其长时间极限下空间轮廓由非可归一化平稳密度控制，时间涨落由Mittag-Leffler随机振幅支配，并给出占据时间的Darling-Kac涨落及幂次观测量的平方Bessel型极限，为无穷遍历理论提供统一框架。
Introduction to Infinite Dimensional Statistics and Applications
本文介绍了无穷维统计的基础知识，涵盖 Hilbert 空间中的随机过程、Karhunen-Loève 展开及 EnKF 等主题，并提供了详细证明以供参考。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。