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Asymptotic Burnside laws 渐近伯恩塞德定律
arXiv - MATH - Probability Pub Date : 2024-09-15 DOI: arxiv-2409.09630
Gil Goffer, Be'eri Greenfeld, Alexander Yu. Olshanskii
{"title":"Asymptotic Burnside laws","authors":"Gil Goffer, Be'eri Greenfeld, Alexander Yu. Olshanskii","doi":"arxiv-2409.09630","DOIUrl":"https://doi.org/arxiv-2409.09630","url":null,"abstract":"We construct novel examples of finitely generated groups that exhibit\u0000seemingly-contradicting probabilistic behaviors with respect to Burnside laws.\u0000We construct a finitely generated group that satisfies a Burnside law, namely a\u0000law of the form $x^n=1$, with limit probability 1 with respect to uniform\u0000measures on balls in its Cayley graph and under every lazy non-degenerate\u0000random walk, while containing a free subgroup. We show that the limit\u0000probability of satisfying a Burnside law is highly sensitive to the choice of\u0000generating set, by providing a group for which this probability is $0$ for one\u0000generating set and $1$ for another. Furthermore, we construct groups that\u0000satisfy Burnside laws of two co-prime exponents with probability 1. Finally, we\u0000present a finitely generated group for which every real number in the interval\u0000$[0,1]$ appears as a partial limit of the probability sequence of Burnside law\u0000satisfaction, both for uniform measures on Cayley balls and for random walks. Our results resolve several open questions posed by Amir, Blachar,\u0000Gerasimova, and Kozma. The techniques employed in this work draw upon geometric\u0000analysis of relations in groups, information-theoretic coding theory on groups,\u0000and combinatorial and probabilistic methods.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262510","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
Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes 高斯、泊松和拉德马赫过程的非均匀贝里--埃森边界
arXiv - MATH - Probability Pub Date : 2024-09-14 DOI: arxiv-2409.09439
Marius Butzek, Peter Eichelsbacher
{"title":"Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes","authors":"Marius Butzek, Peter Eichelsbacher","doi":"arxiv-2409.09439","DOIUrl":"https://doi.org/arxiv-2409.09439","url":null,"abstract":"In this paper we obtain non-uniform Berry-Esseen bounds for normal\u0000approximations by the Malliavin-Stein method. The techniques rely on a detailed\u0000analysis of the solutions of Stein's equations and will be applied to\u0000functionals of a Gaussian process like multiple Wiener-It^o integrals, to\u0000Poisson functionals as well as to the Rademacher chaos expansion. Second-order\u0000Poincar'e inequalities for normal approximation of these functionals are\u0000connected with non-uniform bounds as well. As applications, elements living\u0000inside a fixed Wiener chaos associated with an isonormal Gaussian process, like\u0000the discretized version of the quadratic variation of a fractional Brownian\u0000motion, are considered. Moreover we consider subgraph counts in random\u0000geometric graphs as an example of Poisson $U$-statistics, as well as subgraph\u0000counts in the ErdH{o}s-R'enyi random graph and infinite weighted 2-runs as\u0000examples of functionals of Rademacher variables.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"209 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262500","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
Universal generalized functionals and finitely absolutely continuous measures on Banach spaces 巴拿赫空间上的通用广义函数和有限绝对连续度量
arXiv - MATH - Probability Pub Date : 2024-09-14 DOI: arxiv-2409.09303
A. A. Dorogovtsev, Naoufel Salhi
{"title":"Universal generalized functionals and finitely absolutely continuous measures on Banach spaces","authors":"A. A. Dorogovtsev, Naoufel Salhi","doi":"arxiv-2409.09303","DOIUrl":"https://doi.org/arxiv-2409.09303","url":null,"abstract":"In this paper we collect several examples of convergence of functions of\u0000random processes to generalized functionals of those processes. We remark that\u0000the limit is always finitely absolutely continuous with respect to Wiener\u0000measure. We try to unify those examples in terms of convergence of probability\u0000measures in Banach spaces. The key notion is the condition of uniform finite\u0000absolute continuity.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262503","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
Long distance propagation of wave beams in paraxial regime 波束在准轴向的长距离传播
arXiv - MATH - Probability Pub Date : 2024-09-14 DOI: arxiv-2409.09514
Guillaume Bal, Anjali Nair
{"title":"Long distance propagation of wave beams in paraxial regime","authors":"Guillaume Bal, Anjali Nair","doi":"arxiv-2409.09514","DOIUrl":"https://doi.org/arxiv-2409.09514","url":null,"abstract":"This paper concerns the propagation of high frequency wave-beams in highly\u0000turbulent atmospheres. Using a paraxial model of wave propagation, we show in\u0000the long-distance weak-coupling regime that the wavefields are approximately\u0000described by a complex Gaussian field whose scintillation index is unity. This\u0000provides a model of the speckle formation observed in many practical settings.\u0000The main step of the derivation consists in showing that closed-form moment\u0000equations in the It^o-Schr\"odinger regime are still approximately satisfied\u0000in the paraxial regime. The rest of the proof is then an extension of results\u0000derived in [Bal, G. and Nair, A., arXiv:2402.17107.]","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262704","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
A study on the $F$-distribution motivated by Chvátal's theorem 由 Chvátal 定理引发的 F$ 分布研究
arXiv - MATH - Probability Pub Date : 2024-09-14 DOI: arxiv-2409.09420
Qianqian Zhou, Peng Lu, Zechun Hu
{"title":"A study on the $F$-distribution motivated by Chvátal's theorem","authors":"Qianqian Zhou, Peng Lu, Zechun Hu","doi":"arxiv-2409.09420","DOIUrl":"https://doi.org/arxiv-2409.09420","url":null,"abstract":"Let $X_{d_1, d_2}$ be an $F$-random variable with parameters $d_1$ and $d_2,$\u0000and expectation $E[X_{d_1, d_2}]$. In this paper, for any $kappa>0,$ we\u0000investigate the infimum value of the probability $P(X_{d_1, d_2}leq kappa\u0000E[X_{d_1, d_2}])$. Our motivation comes from Chv'{a}tal's theorem on the\u0000binomial distribution.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"91 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262501","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
The (n,k) game with heterogeneous agents 具有异质代理的（n,k）博弈
arXiv - MATH - Probability Pub Date : 2024-09-14 DOI: arxiv-2409.09364
Hsin-Lun Li
{"title":"The (n,k) game with heterogeneous agents","authors":"Hsin-Lun Li","doi":"arxiv-2409.09364","DOIUrl":"https://doi.org/arxiv-2409.09364","url":null,"abstract":"The ((n,k)) game models a group of (n) individuals with binary opinions,\u0000say 1 and 0, where a decision is made if at least (k) individuals hold\u0000opinion 1. This paper explores the dynamics of the game with heterogeneous\u0000agents under both synchronous and asynchronous settings. We consider various\u0000agent types, including consentors, who always hold opinion 1, rejectors, who\u0000consistently hold opinion 0, random followers, who imitate one of their social\u0000neighbors at random, and majority followers, who adopt the majority opinion\u0000among their social neighbors. We investigate the likelihood of a decision being\u0000made in finite time. In circumstances where a decision cannot almost surely be\u0000made in finite time, we derive a nontrivial bound to offer insight into the\u0000probability of a decision being made in finite time.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262504","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
An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots 具有 i.i.d. 根的随机多项式的高斯-卢卡斯定理的渐近改进
arXiv - MATH - Probability Pub Date : 2024-09-14 DOI: arxiv-2409.09538
Sean O'Rourke, Noah Williams
{"title":"An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots","authors":"Sean O'Rourke, Noah Williams","doi":"arxiv-2409.09538","DOIUrl":"https://doi.org/arxiv-2409.09538","url":null,"abstract":"If $p:mathbb{C} to mathbb{C}$ is a non-constant polynomial, the\u0000Gauss--Lucas theorem asserts that its critical points are contained in the\u0000convex hull of its roots. We consider the case when $p$ is a random polynomial\u0000of degree $n$ with roots chosen independently from a radially symmetric,\u0000compactly supported probably measure $mu$ in the complex plane. We show that\u0000the largest (in magnitude) critical points are closely paired with the largest\u0000roots of $p$. This allows us to compute the asymptotic fluctuations of the\u0000largest critical points as the degree $n$ tends to infinity. We show that the\u0000limiting distribution of the fluctuations is described by either a Gaussian\u0000distribution or a heavy-tailed stable distribution, depending on the behavior\u0000of $mu$ near the edge of its support. As a corollary, we obtain an asymptotic\u0000refinement to the Gauss--Lucas theorem for random polynomials.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262498","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
Asymptotic analysis in problems with fractional processes 分数过程问题的渐近分析
arXiv - MATH - Probability Pub Date : 2024-09-14 DOI: arxiv-2409.09377
P. Chigansky, M. Kleptsyna
{"title":"Asymptotic analysis in problems with fractional processes","authors":"P. Chigansky, M. Kleptsyna","doi":"arxiv-2409.09377","DOIUrl":"https://doi.org/arxiv-2409.09377","url":null,"abstract":"Some problems in the theory and applications of stochastic processes can be\u0000reduced to solving integral equations. Such equations, however, rarely have\u0000explicit solutions. Useful information can be obtained by means of their\u0000asymptotic analysis with respect to relevant parameters. This paper is a brief\u0000survey of some recent progress in the study of such equations related to\u0000processes with fractional covariance structure.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262502","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
Characterizations of $A_infty$ Weights in Ergodic Theory 遍历理论中的 $A_infty$ 权重特征
arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.08896
Wei Chen, Jingyi Wang
{"title":"Characterizations of $A_infty$ Weights in Ergodic Theory","authors":"Wei Chen, Jingyi Wang","doi":"arxiv-2409.08896","DOIUrl":"https://doi.org/arxiv-2409.08896","url":null,"abstract":"We establish a discrete weighted version of Calder'{o}n-Zygmund\u0000decomposition from the perspective of dyadic grid in ergodic theory. Based on\u0000the decomposition, we study discrete $A_infty$ weights. First,\u0000characterizations of the reverse H\"{o}lder's inequality and their extensions\u0000are obtained. Second, the properties of $A_infty$ are given, specifically\u0000$A_infty$ implies the reverse H\"{o}lder's inequality. Finally, under a\u0000doubling condition on weights, $A_infty$ follows from the reverse H\"{o}lder's\u0000inequality. This means that we obtain equivalent characterizations of\u0000$A_{infty}$. Because $A_{infty}$ implies the doubling condition, it seems\u0000reasonable to assume the condition.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262709","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
Markov chains, CAT(0) cube complexes, and enumeration: monotone paths in a strip mix slowly 马尔可夫链、CAT(0)立方体复合物和枚举：带状混合体中的单调路径缓慢变化
arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.09133
Federico Ardila-Mantilla, Naya Banerjee, Coleson Weir
{"title":"Markov chains, CAT(0) cube complexes, and enumeration: monotone paths in a strip mix slowly","authors":"Federico Ardila-Mantilla, Naya Banerjee, Coleson Weir","doi":"arxiv-2409.09133","DOIUrl":"https://doi.org/arxiv-2409.09133","url":null,"abstract":"We prove that two natural Markov chains on the set of monotone paths in a\u0000strip mix slowly. To do so, we make novel use of the theory of non-positively\u0000curved (CAT(0)) cubical complexes to detect small bottlenecks in many graphs of\u0000combinatorial interest. Along the way, we give a formula for the number c_m(n)\u0000of monotone paths of length n in a strip of height m. In particular we compute\u0000the exponential growth constant of c_m(n) for arbitrary m, generalizing results\u0000of Williams for m=2, 3.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262511","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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