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Large deviation inequalities for the nonlinear unbalanced urn model 非线性不平衡瓮模型的大偏差不等式
Pub Date : 2024-09-12 DOI: arxiv-2409.07800
Jianan Shi, Zhenhong Yu, Yu Miao
In the present paper, we consider the two-color nonlinear unbalanced urnmodel, under a drawing rule reinforced by an $mathbb{R}^+$-valued concavefunction and an unbalanced replacement matrix. The large deviation inequalitiesfor the nonlinear unbalanced urn model are established by using the stochasticapproximation theory. As an auxiliary theory, we give a specific largedeviation inequality for a general stochastic approximation algorithm.
本文考虑了在$mathbb{R}^+$值凹函数和不平衡替换矩阵强化的抽签规则下的双色非线性不平衡瓮模型。我们利用随机逼近理论建立了非线性不平衡瓮模型的大偏差不等式。作为辅助理论,我们给出了一般随机逼近算法的具体大偏差不等式。
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引用次数: 0
Limit Profile for the Bernoulli--Laplace Urn 伯努利-拉普拉斯瓮的极限轮廓
Pub Date : 2024-09-12 DOI: arxiv-2409.07900
Sam Olesker-Taylor, Dominik Schmid
We analyse the convergence to equilibrium of the Bernoulli--Laplace urnmodel: initially, one urn contains $k$ red balls and a second $n-k$ blue balls;in each step, a pair of balls is chosen uniform and their locations areswitched. Cutoff is known to occur at $tfrac12 n log min{k, sqrt n}$ withwindow order $n$ whenever $1 ll k le tfrac12 n$. We refine this bydetermining the limit profile: a function $Phi$ such that [ d_mathsf{TV}bigl( tfrac12 n log min{k, sqrt n} + theta n bigr) to Phi(theta) quadtext{as}quad n to infty quadtext{for all}quad theta in mathbb R. ] Our main technical contribution, of independentinterest, approximates a rescaled chain by a diffusion on $mathbb R$ when $kgg sqrt n$, and uses its explicit law as a Gaussian process.
我们分析了伯努利--拉普拉斯瓮模型向均衡收敛的过程:最初,一个瓮包含 $k$ 红球,第二个瓮包含 $n-k$ 蓝球;在每一步中,均匀地选择一对球,并切换它们的位置。众所周知,当 1 ll k le tfrac12 n$ 时,截止点会出现在 $tfrac12 n log min{k, sqrt n}$,窗口阶数为 $n$。我们通过确定极限轮廓来完善这一点:a function $Phi$ such that [ d_mathsf{TV}bigl( tfrac12 n log min{k, sqrt n} + theta n bigr) to Phi(theta) quadtext{as}quad n to infty quadtext{for all}quad theta in mathbb R.]我们的主要技术贡献是,当 $kgg sqrt n$ 时,用 $mathbb R$ 上的扩散来近似一个重标度链,并将其显式规律作为一个高斯过程。
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引用次数: 0
Quantitative periodic homogenization for symmetric non-local stable-like operators 对称非局部稳定类算子的定量周期同质化
Pub Date : 2024-09-12 DOI: arxiv-2409.08120
Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang
Homogenization for non-local operators in periodic environments has beenstudied intensively. So far, these works are mainly devoted to the qualitativeresults, that is, to determine explicitly the operators in the limit. To thebest of authors' knowledge, there is no result concerning the convergence ratesof the homogenization for stable-like operators in periodic environments. Inthis paper, we establish a quantitative homogenization result for symmetric$alpha$-stable-like operators on $R^d$ with periodic coefficients. Inparticular, we show that the convergence rate for the solutions of associatedDirichlet problems on a bounded domain $D$ is of order $$varepsilon^{(2-alpha)/2}I_{{alphain(1,2)}}+varepsilon^{alpha/2}I_{{alphain (0,1)}}+varepsilon^{1/2}|loge|^2I_{{alpha=1}}, $$ while, when the solution to the equation in thelimit is in $C^2_c(D)$, the convergence rate becomes $$ varepsilon^{2-alpha}I_{{alphain(1,2)}}+varepsilon^{alpha}I_{{alphain (0,1)}}+varepsilon |loge|^2I_{{alpha=1}}. $$ This indicates that the boundary decay behaviors ofthe solution to the equation in the limit affects the convergence rate in thehomogenization.
人们对周期环境中的非局部算子的均质化进行了深入研究。迄今为止,这些研究主要致力于定性结果,即明确确定极限中的算子。据作者所知,目前还没有关于周期环境中稳定类算子同质化收敛率的结果。在本文中,我们建立了具有周期性系数的 $R^d$ 上对称$alpha$稳定类算子的定量同质化结果。特别是,我们证明了在有界域 $D$ 上相关德里赫特问题解的收敛速率为 $$varepsilon^{(2-alpha)/2}I_{{alphain(1、2)}}+varepsilon^{alpha/2}I_{{alphain (0,1)}}+varepsilon^{1/2}|loge|^2I_{{alpha=1}},$$ 而当极限中方程的解位于 $C^2_c(D)$ 时,收敛速率变为 $$ varepsilon^{2-alpha}I_{alphain(1、2)}}+varepsilon^{alpha}I_{{alphain (0,1)}}+varepsilon |loge|^2I_{{alpha=1}}.$$ 这表明方程解在极限时的边界衰减行为会影响均质化的收敛速度。
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引用次数: 0
On a class of exponential changes of measure for stochastic PDEs 论随机 PDE 的一类指数量变
Pub Date : 2024-09-12 DOI: arxiv-2409.08057
Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart
Given a mild solution $X$ to a semilinear stochastic partial differentialequation (SPDE), we consider an exponential change of measure based on itsinfinitesimal generator $L$, defined in the topology of bounded pointwiseconvergence. The changed measure $mathbb{P}^h$ depends on the choice of afunction $h$ in the domain of $L$. In our main result, we derive conditions on$h$ for which the change of measure is of Girsanov-type. The process $X$ under$mathbb{P}^h$ is then shown to be a mild solution to another SPDE with anextra additive drift-term. We illustrate how different choices of $h$ impactthe law of $X$ under $mathbb{P}^h$ in selected applications. These include thederivation of an infinite-dimensional diffusion bridge as well as theintroduction of guided processes for SPDEs, generalizing results known forfinite-dimensional diffusion processes to the infinite-dimensional case.
给定半线性随机偏微分方程(SPDE)的温和解$X$,我们考虑基于其无限小生成器$L$的指数变化度量,该度量在有界点顺收敛拓扑中定义。变化后的度量 $mathbb{P}^h$ 取决于在 $L$ 的域中选择一个函数 $h$。在我们的主要结果中,我们推导出了关于$h$ 的条件,在这些条件下,度量的变化属于吉尔萨诺夫类型。然后,我们证明了$X$ 在$mathbb{P}^h$ 下的过程是另一个具有额外加漂移项的 SPDE 的温和解。我们在选定的应用中说明了不同的 $h$ 选择如何影响 $X$ 在 $mathbb{P}^h$ 下的规律。这些应用包括无穷维扩散桥的衍生,以及引入 SPDE 的引导过程,将已知的无穷维扩散过程的结果推广到无穷维情况。
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引用次数: 0
Khintchine dichotomy for self-similar measures 自相似度量的钦钦二分法
Pub Date : 2024-09-12 DOI: arxiv-2409.08061
Timothée Bénard, Weikun He, Han Zhang
We establish the analogue of Khintchine's theorem for all self-similarprobability measures on the real line. When specified to the case of theHausdorff measure on the middle-thirds Cantor set, the result is already newand provides an answer to an old question of Mahler. The proof consists inshowing effective equidistribution in law of expanding upper-triangular randomwalks on $text{SL}_{2}(mathbb{R})/text{SL}_{2}(mathbb{Z})$, a result ofindependent interest.
我们为实线上的所有自相似概率度量建立了Khintchine定理的类似物。当把这一定理应用于中三康托尔集上的豪斯多夫度量时,它已经是一个新结果,并为马勒的一个老问题提供了答案。证明包括在$text{SL}_{2}(mathbb{R})/text{SL}_{2}(mathbb{Z})$上的扩展上三角随机游走法则中显示有效的等差数列,这是一个具有独立意义的结果。
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引用次数: 0
Random walks with stochastic resetting in complex networks: a discrete time approach 复杂网络中的随机重置随机漫步:一种离散时间方法
Pub Date : 2024-09-12 DOI: arxiv-2409.08394
Thomas M. Michelitsch, Giuseppe D'Onofrio, Federico Polito, Alejandro P. Riascos
We consider a discrete-time Markovian random walk with resets on a connectedundirected network. The resets, in which the walker is relocated to randomlychosen nodes, are governed by an independent discrete-time renewal process.Some nodes of the network are target nodes, and we focus on the statistics offirst hitting of these nodes. In the non-Markov case of the renewal process, weconsider both light- and fat-tailed inter-reset distributions. We derive thepropagator matrix in terms of discrete backward recurrence time PDFs and in thelight-tailed case we show the existence of a non-equilibrium steady state. Inorder to tackle the non-Markov scenario, we derive a defective propagatormatrix which describes an auxiliary walk characterized by killing the walker assoon as it hits target nodes. This propagator provides the information on themean first passage statistics to the target nodes. We establish sufficientconditions for ergodicity of the walk under resetting. Furthermore, we discussa generic resetting mechanism for which the walk is non-ergodic. Finally, weanalyze inter-reset time distributions with infinite mean where we focus on theSibuya case. We apply these results to study the mean first passage times forMarkovian and non-Markovian (Sibuya) renewal resetting protocols inrealizations of Watts-Strogatz and Barab'asi-Albert random graphs. We show nontrivial behavior of the dependence of the mean first passage time on theproportions of the relocation nodes, target nodes and of the resetting rates.It turns out that, in the large-world case of the Watts-Strogatz graph, theefficiency of a random searcher particularly benefits from the presence ofresets.
我们考虑的是在连通的有向网络上进行重置的离散时间马尔可夫随机行走。网络中的某些节点是目标节点,我们将重点放在这些节点的首次命中统计上。在更新过程的非马尔可夫情况下,我们考虑了轻尾和胖尾的重置间分布。我们根据离散后向递推时间 PDF 推导出传播矩阵,并在光尾情况下证明了非平衡稳态的存在。为了解决非马尔可夫情况,我们推导出了一个有缺陷的传播矩阵,它描述了一种辅助行走,其特征是当行走者到达目标节点时立即杀死行走者。该传播器为目标节点提供了关于主题的首次通过统计信息。我们建立了在重置条件下行走的遍历性的充分条件。此外,我们还讨论了一般的重置机制,在这种机制下,行走是非遍历性的。最后,我们分析了具有无限均值的重置间时间分布,并将重点放在西布亚(Sibuya)情况上。我们应用这些结果研究了在瓦特-斯特罗加茨和巴拉布-阿西-阿尔伯特随机图的现实化中马尔可夫和非马尔可夫(西布亚)更新重置协议的平均首次通过时间。我们展示了平均首次通过时间与重新定位节点、目标节点和重置率的比例之间非同一般的依赖关系。事实证明,在瓦茨-斯特罗加茨图的大世界情形中,随机搜索器的效率尤其受益于重置的存在。
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引用次数: 0
Upper tails for arithmetic progressions revisited 再论算术级数的上尾数
Pub Date : 2024-09-12 DOI: arxiv-2409.08383
Matan Harel, Frank Mousset, Wojciech Samotij
Let $X$ be the number of $k$-term arithmetic progressions contained in the$p$-biased random subset of the first $N$ positive integers. We giveasymptotically sharp estimates on the logarithmic upper-tail probability $logPr(X ge E[X] + t)$ for all $Omega(N^{-2/k}) le p ll 1$ and all $t ggsqrt{Var(X)}$, excluding only a few boundary cases. In particular, we showthat the space of parameters $(p,t)$ is partitioned into threephenomenologically distinct regions, where the upper-tail probabilities eitherresemble those of Gaussian or Poisson random variables, or are naturallydescribed by the probability of appearance of a small set that contains nearlyall of the excess $t$ progressions. We employ a variety of tools fromprobability theory, including classical tilting arguments and martingaleconcentration inequalities. However, the main technical innovation is acombinatorial result that establishes a stronger version of `entropicstability' for sets with rich arithmetic structure.
让 $X$ 是前 $N$ 正整数的 $p$ 偏随机子集中包含的 $k$ 期算术级数的数目。对于所有$Omega(N^{-2/k}) le p ll 1$和所有$t ggsqrt{Var(X)}$,我们给出了对数上尾概率$logPr(X ge E[X] +t)$的渐近尖锐估计,仅排除了一些边界情况。我们特别指出,参数 $(p,t)$ 的空间被划分为三个现象学上截然不同的区域,其中上尾概率要么类似于高斯或泊松随机变量的概率,要么自然地由一个小集合的出现概率来描述,而这个小集合几乎包含了所有多余的 $t$ 级数。我们采用了概率论中的各种工具,包括经典的倾斜论证和马氏集中不等式。然而,主要的技术创新是一个组合结果,它为具有丰富算术结构的集合建立了一个更强版本的 "熵稳定性"。
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引用次数: 0
Rapid mixing of the flip chain over non-crossing spanning trees 非交叉生成树上翻转链的快速混合
Pub Date : 2024-09-12 DOI: arxiv-2409.07892
Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, Mark Jerrum, Jiaheng Wang
We show that the flip chain for non-crossing spanning trees of $n+1$ pointsin convex position mixes in time $O(n^8log n)$.
我们证明,在凸位置上的 $n+1$ 点的非交叉生成树的翻转链混合时间为 $O(n^8log n)$。
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引用次数: 0
Dimensions of harmonic measures on non-autonomous Cantor sets 非自治康托尔集上的调和度量维数
Pub Date : 2024-09-12 DOI: arxiv-2409.08019
Athanasios Batakis, Guillaume Havard
We consider Non Autonomous Conformal Iterative Function Systems (NACIFS) andtheir limit set. Our main concern is harmonic measure and its dimensions :Hausdorff and Packing. We prove that this two dimensions are continuous underperturbations and that they verify Bowen's and Manning's type formulas. Inorder to do so we prove general results about measures, and more generallyabout positive functionals, defined on a symbolic space, developing tools fromthermodynamical formalism in a non-autonomous setting.
我们考虑非自治共形迭代函数系统(NACIFS)及其极限集。我们主要关注调和度量及其维度:Hausdorff 和 Packing。我们证明这两个维度在扰动下是连续的,而且它们验证了鲍温和曼宁类型公式。为此,我们证明了定义在符号空间上的度量的一般结果,以及更广义的正函数的一般结果,并在非自治环境中开发了热力学形式主义的工具。
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引用次数: 0
Sylvester's problem for random walks and bridges 随机漫步和桥梁的西尔维斯特问题
Pub Date : 2024-09-12 DOI: arxiv-2409.07927
Hugo Panzo
Consider a random walk in $mathbb{R}^d$ that starts at the origin and whoseincrement distribution assigns zero probability to any affine hyperplane. Wesolve Sylvester's problem for these random walks by showing that theprobability that the first $d+2$ steps of the walk are in convex position isequal to $1-frac{2}{(d+1)!}$. The analogous result also holds for randombridges of length $d+2$, so long as the joint increment distribution isexchangeable.
考虑$mathbb{R}^d$中的随机行走,它从原点开始,其增量分布赋予任何仿射超平面的概率为零。通过证明行走的前 $d+2$ 步处于凸位置的概率等于$1-frac{2}{(d+1)!}$,解决这些随机行走的西尔维斯特问题。只要联合增量分布是可交换的,类似的结果对于长度为 $d+2$ 的随机走廊也是成立的。
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引用次数: 0
期刊
arXiv - MATH - Probability
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