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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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.07800
Jianan Shi, Zhenhong Yu, Yu Miao
{"title":"Large deviation inequalities for the nonlinear unbalanced urn model","authors":"Jianan Shi, Zhenhong Yu, Yu Miao","doi":"arxiv-2409.07800","DOIUrl":"https://doi.org/arxiv-2409.07800","url":null,"abstract":"In the present paper, we consider the two-color nonlinear unbalanced urn\u0000model, under a drawing rule reinforced by an $mathbb{R}^+$-valued concave\u0000function and an unbalanced replacement matrix. The large deviation inequalities\u0000for the nonlinear unbalanced urn model are established by using the stochastic\u0000approximation theory. As an auxiliary theory, we give a specific large\u0000deviation inequality for a general stochastic approximation algorithm.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.07900
Sam Olesker-Taylor, Dominik Schmid
{"title":"Limit Profile for the Bernoulli--Laplace Urn","authors":"Sam Olesker-Taylor, Dominik Schmid","doi":"arxiv-2409.07900","DOIUrl":"https://doi.org/arxiv-2409.07900","url":null,"abstract":"We analyse the convergence to equilibrium of the Bernoulli--Laplace urn\u0000model: initially, one urn contains $k$ red balls and a second $n-k$ blue balls;\u0000in each step, a pair of balls is chosen uniform and their locations are\u0000switched. Cutoff is known to occur at $tfrac12 n log min{k, sqrt n}$ with\u0000window order $n$ whenever $1 ll k le tfrac12 n$. We refine this by\u0000determining 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 independent\u0000interest, approximates a rescaled chain by a diffusion on $mathbb R$ when $k\u0000gg sqrt n$, and uses its explicit law as a Gaussian process.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.08120
Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang
{"title":"Quantitative periodic homogenization for symmetric non-local stable-like operators","authors":"Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang","doi":"arxiv-2409.08120","DOIUrl":"https://doi.org/arxiv-2409.08120","url":null,"abstract":"Homogenization for non-local operators in periodic environments has been\u0000studied intensively. So far, these works are mainly devoted to the qualitative\u0000results, that is, to determine explicitly the operators in the limit. To the\u0000best of authors' knowledge, there is no result concerning the convergence rates\u0000of the homogenization for stable-like operators in periodic environments. In\u0000this paper, we establish a quantitative homogenization result for symmetric\u0000$alpha$-stable-like operators on $R^d$ with periodic coefficients. In\u0000particular, we show that the convergence rate for the solutions of associated\u0000Dirichlet problems on a bounded domain $D$ is of order $$\u0000varepsilon^{(2-alpha)/2}I_{{alphain\u0000(1,2)}}+varepsilon^{alpha/2}I_{{alphain (0,1)}}+varepsilon^{1/2}|log\u0000e|^2I_{{alpha=1}}, $$ while, when the solution to the equation in the\u0000limit is in $C^2_c(D)$, the convergence rate becomes $$ varepsilon^{2-alpha}I_{{alphain\u0000(1,2)}}+varepsilon^{alpha}I_{{alphain (0,1)}}+varepsilon |log\u0000e|^2I_{{alpha=1}}. $$ This indicates that the boundary decay behaviors of\u0000the solution to the equation in the limit affects the convergence rate in the\u0000homogenization.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.08057
Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart
{"title":"On a class of exponential changes of measure for stochastic PDEs","authors":"Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart","doi":"arxiv-2409.08057","DOIUrl":"https://doi.org/arxiv-2409.08057","url":null,"abstract":"Given a mild solution $X$ to a semilinear stochastic partial differential\u0000equation (SPDE), we consider an exponential change of measure based on its\u0000infinitesimal generator $L$, defined in the topology of bounded pointwise\u0000convergence. The changed measure $mathbb{P}^h$ depends on the choice of a\u0000function $h$ in the domain of $L$. In our main result, we derive conditions on\u0000$h$ for which the change of measure is of Girsanov-type. The process $X$ under\u0000$mathbb{P}^h$ is then shown to be a mild solution to another SPDE with an\u0000extra additive drift-term. We illustrate how different choices of $h$ impact\u0000the law of $X$ under $mathbb{P}^h$ in selected applications. These include the\u0000derivation of an infinite-dimensional diffusion bridge as well as the\u0000introduction of guided processes for SPDEs, generalizing results known for\u0000finite-dimensional diffusion processes to the infinite-dimensional case.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Khintchine dichotomy for self-similar measures 自相似度量的钦钦二分法
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.08061
Timothée Bénard, Weikun He, Han Zhang
{"title":"Khintchine dichotomy for self-similar measures","authors":"Timothée Bénard, Weikun He, Han Zhang","doi":"arxiv-2409.08061","DOIUrl":"https://doi.org/arxiv-2409.08061","url":null,"abstract":"We establish the analogue of Khintchine's theorem for all self-similar\u0000probability measures on the real line. When specified to the case of the\u0000Hausdorff measure on the middle-thirds Cantor set, the result is already new\u0000and provides an answer to an old question of Mahler. The proof consists in\u0000showing effective equidistribution in law of expanding upper-triangular random\u0000walks on $text{SL}_{2}(mathbb{R})/text{SL}_{2}(mathbb{Z})$, a result of\u0000independent interest.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.08394
Thomas M. Michelitsch, Giuseppe D'Onofrio, Federico Polito, Alejandro P. Riascos
{"title":"Random walks with stochastic resetting in complex networks: a discrete time approach","authors":"Thomas M. Michelitsch, Giuseppe D'Onofrio, Federico Polito, Alejandro P. Riascos","doi":"arxiv-2409.08394","DOIUrl":"https://doi.org/arxiv-2409.08394","url":null,"abstract":"We consider a discrete-time Markovian random walk with resets on a connected\u0000undirected network. The resets, in which the walker is relocated to randomly\u0000chosen nodes, are governed by an independent discrete-time renewal process.\u0000Some nodes of the network are target nodes, and we focus on the statistics of\u0000first hitting of these nodes. In the non-Markov case of the renewal process, we\u0000consider both light- and fat-tailed inter-reset distributions. We derive the\u0000propagator matrix in terms of discrete backward recurrence time PDFs and in the\u0000light-tailed case we show the existence of a non-equilibrium steady state. In\u0000order to tackle the non-Markov scenario, we derive a defective propagator\u0000matrix which describes an auxiliary walk characterized by killing the walker as\u0000soon as it hits target nodes. This propagator provides the information on the\u0000mean first passage statistics to the target nodes. We establish sufficient\u0000conditions for ergodicity of the walk under resetting. Furthermore, we discuss\u0000a generic resetting mechanism for which the walk is non-ergodic. Finally, we\u0000analyze inter-reset time distributions with infinite mean where we focus on the\u0000Sibuya case. We apply these results to study the mean first passage times for\u0000Markovian and non-Markovian (Sibuya) renewal resetting protocols in\u0000realizations of Watts-Strogatz and Barab'asi-Albert random graphs. We show non\u0000trivial behavior of the dependence of the mean first passage time on the\u0000proportions of the relocation nodes, target nodes and of the resetting rates.\u0000It turns out that, in the large-world case of the Watts-Strogatz graph, the\u0000efficiency of a random searcher particularly benefits from the presence of\u0000resets.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Upper tails for arithmetic progressions revisited 再论算术级数的上尾数
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.08383
Matan Harel, Frank Mousset, Wojciech Samotij
{"title":"Upper tails for arithmetic progressions revisited","authors":"Matan Harel, Frank Mousset, Wojciech Samotij","doi":"arxiv-2409.08383","DOIUrl":"https://doi.org/arxiv-2409.08383","url":null,"abstract":"Let $X$ be the number of $k$-term arithmetic progressions contained in the\u0000$p$-biased random subset of the first $N$ positive integers. We give\u0000asymptotically sharp estimates on the logarithmic upper-tail probability $log\u0000Pr(X ge E[X] + t)$ for all $Omega(N^{-2/k}) le p ll 1$ and all $t gg\u0000sqrt{Var(X)}$, excluding only a few boundary cases. In particular, we show\u0000that the space of parameters $(p,t)$ is partitioned into three\u0000phenomenologically distinct regions, where the upper-tail probabilities either\u0000resemble those of Gaussian or Poisson random variables, or are naturally\u0000described by the probability of appearance of a small set that contains nearly\u0000all of the excess $t$ progressions. We employ a variety of tools from\u0000probability theory, including classical tilting arguments and martingale\u0000concentration inequalities. However, the main technical innovation is a\u0000combinatorial result that establishes a stronger version of `entropic\u0000stability' for sets with rich arithmetic structure.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.07892
Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, Mark Jerrum, Jiaheng Wang
{"title":"Rapid mixing of the flip chain over non-crossing spanning trees","authors":"Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, Mark Jerrum, Jiaheng Wang","doi":"arxiv-2409.07892","DOIUrl":"https://doi.org/arxiv-2409.07892","url":null,"abstract":"We show that the flip chain for non-crossing spanning trees of $n+1$ points\u0000in convex position mixes in time $O(n^8log n)$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.08019
Athanasios Batakis, Guillaume Havard
{"title":"Dimensions of harmonic measures on non-autonomous Cantor sets","authors":"Athanasios Batakis, Guillaume Havard","doi":"arxiv-2409.08019","DOIUrl":"https://doi.org/arxiv-2409.08019","url":null,"abstract":"We consider Non Autonomous Conformal Iterative Function Systems (NACIFS) and\u0000their limit set. Our main concern is harmonic measure and its dimensions :\u0000Hausdorff and Packing. We prove that this two dimensions are continuous under\u0000perturbations and that they verify Bowen's and Manning's type formulas. In\u0000order to do so we prove general results about measures, and more generally\u0000about positive functionals, defined on a symbolic space, developing tools from\u0000thermodynamical formalism in a non-autonomous setting.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.07927
Hugo Panzo
{"title":"Sylvester's problem for random walks and bridges","authors":"Hugo Panzo","doi":"arxiv-2409.07927","DOIUrl":"https://doi.org/arxiv-2409.07927","url":null,"abstract":"Consider a random walk in $mathbb{R}^d$ that starts at the origin and whose\u0000increment distribution assigns zero probability to any affine hyperplane. We\u0000solve Sylvester's problem for these random walks by showing that the\u0000probability that the first $d+2$ steps of the walk are in convex position is\u0000equal to $1-frac{2}{(d+1)!}$. The analogous result also holds for random\u0000bridges of length $d+2$, so long as the joint increment distribution is\u0000exchangeable.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"88 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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