Gil Goffer, Be'eri Greenfeld, Alexander Yu. Olshanskii
We construct novel examples of finitely generated groups that exhibit seemingly-contradicting probabilistic behaviors with respect to Burnside laws. We construct a finitely generated group that satisfies a Burnside law, namely a law of the form $x^n=1$, with limit probability 1 with respect to uniform measures on balls in its Cayley graph and under every lazy non-degenerate random walk, while containing a free subgroup. We show that the limit probability of satisfying a Burnside law is highly sensitive to the choice of generating set, by providing a group for which this probability is $0$ for one generating set and $1$ for another. Furthermore, we construct groups that satisfy Burnside laws of two co-prime exponents with probability 1. Finally, we present a finitely generated group for which every real number in the interval $[0,1]$ appears as a partial limit of the probability sequence of Burnside law satisfaction, both for uniform measures on Cayley balls and for random walks. Our results resolve several open questions posed by Amir, Blachar, Gerasimova, and Kozma. The techniques employed in this work draw upon geometric analysis of relations in groups, information-theoretic coding theory on groups, and combinatorial and probabilistic methods.
{"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}
In this paper we obtain non-uniform Berry-Esseen bounds for normal approximations by the Malliavin-Stein method. The techniques rely on a detailed analysis of the solutions of Stein's equations and will be applied to functionals of a Gaussian process like multiple Wiener-It^o integrals, to Poisson functionals as well as to the Rademacher chaos expansion. Second-order Poincar'e inequalities for normal approximation of these functionals are connected with non-uniform bounds as well. As applications, elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process, like the discretized version of the quadratic variation of a fractional Brownian motion, are considered. Moreover we consider subgraph counts in random geometric graphs as an example of Poisson $U$-statistics, as well as subgraph counts in the ErdH{o}s-R'enyi random graph and infinite weighted 2-runs as examples of functionals of Rademacher variables.
{"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}
In this paper we collect several examples of convergence of functions of random processes to generalized functionals of those processes. We remark that the limit is always finitely absolutely continuous with respect to Wiener measure. We try to unify those examples in terms of convergence of probability measures in Banach spaces. The key notion is the condition of uniform finite absolute continuity.
{"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}
This paper concerns the propagation of high frequency wave-beams in highly turbulent atmospheres. Using a paraxial model of wave propagation, we show in the long-distance weak-coupling regime that the wavefields are approximately described by a complex Gaussian field whose scintillation index is unity. This provides a model of the speckle formation observed in many practical settings. The main step of the derivation consists in showing that closed-form moment equations in the It^o-Schr"odinger regime are still approximately satisfied in the paraxial regime. The rest of the proof is then an extension of results derived in [Bal, G. and Nair, A., arXiv:2402.17107.]
本文涉及高频波束在高扰动大气中的传播。利用波传播的准轴模型，我们证明了在长距离弱耦合状态下，波场近似由闪烁指数为一的复高斯场来描述。推导的主要步骤包括证明 It^o-Schr"odinger 体系中的闭式矩方程在准轴向体系中仍然近似满足。证明的其余部分是对 [Bal, G. and Nair, A., arXiv:2402.17107.] 中得出的结果的扩展。
{"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}
Let $X_{d_1, d_2}$ be an $F$-random variable with parameters $d_1$ and $d_2,$ and expectation $E[X_{d_1, d_2}]$. In this paper, for any $kappa>0,$ we investigate the infimum value of the probability $P(X_{d_1, d_2}leq kappa E[X_{d_1, d_2}])$. Our motivation comes from Chv'{a}tal's theorem on the binomial distribution.
{"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}
The ((n,k)) game models a group of (n) individuals with binary opinions, say 1 and 0, where a decision is made if at least (k) individuals hold opinion 1. This paper explores the dynamics of the game with heterogeneous agents under both synchronous and asynchronous settings. We consider various agent types, including consentors, who always hold opinion 1, rejectors, who consistently hold opinion 0, random followers, who imitate one of their social neighbors at random, and majority followers, who adopt the majority opinion among their social neighbors. We investigate the likelihood of a decision being made in finite time. In circumstances where a decision cannot almost surely be made in finite time, we derive a nontrivial bound to offer insight into the probability of a decision being made in finite time.
{"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}
If $p:mathbb{C} to mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with roots chosen independently from a radially symmetric, compactly supported probably measure $mu$ in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of $p$. This allows us to compute the asymptotic fluctuations of the largest critical points as the degree $n$ tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of $mu$ near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials.
{"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}
Some problems in the theory and applications of stochastic processes can be reduced to solving integral equations. Such equations, however, rarely have explicit solutions. Useful information can be obtained by means of their asymptotic analysis with respect to relevant parameters. This paper is a brief survey of some recent progress in the study of such equations related to processes with fractional covariance structure.
{"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}
We establish a discrete weighted version of Calder'{o}n-Zygmund decomposition from the perspective of dyadic grid in ergodic theory. Based on the decomposition, we study discrete $A_infty$ weights. First, characterizations of the reverse H"{o}lder's inequality and their extensions are obtained. Second, the properties of $A_infty$ are given, specifically $A_infty$ implies the reverse H"{o}lder's inequality. Finally, under a doubling condition on weights, $A_infty$ follows from the reverse H"{o}lder's inequality. This means that we obtain equivalent characterizations of $A_{infty}$. Because $A_{infty}$ implies the doubling condition, it seems reasonable to assume the condition.
{"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}
We prove that two natural Markov chains on the set of monotone paths in a strip mix slowly. To do so, we make novel use of the theory of non-positively curved (CAT(0)) cubical complexes to detect small bottlenecks in many graphs of combinatorial interest. Along the way, we give a formula for the number c_m(n) of monotone paths of length n in a strip of height m. In particular we compute the exponential growth constant of c_m(n) for arbitrary m, generalizing results of Williams for m=2, 3.
我们证明了星状图中单调路径集合上的两条自然马尔可夫链会缓慢混合。为此，我们新颖地使用了非正曲（CAT(0)）立方复曲面理论，以检测许多具有混杂性的图中的小瓶颈。同时，我们给出了高度为 m 的带状图中长度为 n 的单调路径的数量 c_m(n)的计算公式。特别是，我们计算了任意 m 的 c_m(n)的指数增长常数，推广了威廉姆斯关于 m=2, 3 的结果。
{"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}
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