In the present paper, we consider the two-color nonlinear unbalanced urn model, under a drawing rule reinforced by an $mathbb{R}^+$-valued concave function and an unbalanced replacement matrix. The large deviation inequalities for the nonlinear unbalanced urn model are established by using the stochastic approximation theory. As an auxiliary theory, we give a specific large deviation inequality for a general stochastic approximation algorithm.
{"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":null,"pages":null},"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}
We analyse the convergence to equilibrium of the Bernoulli--Laplace urn model: 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 are switched. Cutoff is known to occur at $tfrac12 n log min{k, sqrt n}$ with window order $n$ whenever $1 ll k le tfrac12 n$. We refine this by determining 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 interest, approximates a rescaled chain by a diffusion on $mathbb R$ when $k gg sqrt n$, 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。
{"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":null,"pages":null},"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}
Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang
Homogenization for non-local operators in periodic environments has been studied intensively. So far, these works are mainly devoted to the qualitative results, that is, to determine explicitly the operators in the limit. To the best of authors' knowledge, there is no result concerning the convergence rates of the homogenization for stable-like operators in periodic environments. In this paper, we establish a quantitative homogenization result for symmetric $alpha$-stable-like operators on $R^d$ with periodic coefficients. In particular, we show that the convergence rate for the solutions of associated Dirichlet problems on a bounded domain $D$ is of order $$ varepsilon^{(2-alpha)/2}I_{{alphain (1,2)}}+varepsilon^{alpha/2}I_{{alphain (0,1)}}+varepsilon^{1/2}|log e|^2I_{{alpha=1}}, $$ while, when the solution to the equation in the limit is in $C^2_c(D)$, the convergence rate becomes $$ varepsilon^{2-alpha}I_{{alphain (1,2)}}+varepsilon^{alpha}I_{{alphain (0,1)}}+varepsilon |log e|^2I_{{alpha=1}}. $$ This indicates that the boundary decay behaviors of the solution to the equation in the limit affects the convergence rate in the homogenization.
{"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":null,"pages":null},"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}
Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart
Given a mild solution $X$ to a semilinear stochastic partial differential equation (SPDE), we consider an exponential change of measure based on its infinitesimal generator $L$, defined in the topology of bounded pointwise convergence. The changed measure $mathbb{P}^h$ depends on the choice of a function $h$ 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 an extra additive drift-term. We illustrate how different choices of $h$ impact the law of $X$ under $mathbb{P}^h$ in selected applications. These include the derivation of an infinite-dimensional diffusion bridge as well as the introduction of guided processes for SPDEs, generalizing results known for finite-dimensional diffusion processes to the infinite-dimensional case.
{"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":null,"pages":null},"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}
We establish the analogue of Khintchine's theorem for all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an answer to an old question of Mahler. The proof consists in showing effective equidistribution in law of expanding upper-triangular random walks on $text{SL}_{2}(mathbb{R})/text{SL}_{2}(mathbb{Z})$, a result of independent interest.
{"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":null,"pages":null},"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}
Thomas M. Michelitsch, Giuseppe D'Onofrio, Federico Polito, Alejandro P. Riascos
We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time PDFs and in the light-tailed case we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barab'asi-Albert random graphs. We show non trivial behavior of the dependence of the mean first passage time on the proportions 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, the efficiency of a random searcher particularly benefits from the presence of resets.
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。
{"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":null,"pages":null},"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}
Let $X$ be the number of $k$-term arithmetic progressions contained in the $p$-biased random subset of the first $N$ positive integers. We give asymptotically sharp estimates on the logarithmic upper-tail probability $log Pr(X ge E[X] + t)$ for all $Omega(N^{-2/k}) le p ll 1$ and all $t gg sqrt{Var(X)}$, excluding only a few boundary cases. In particular, we show that the space of parameters $(p,t)$ is partitioned into three phenomenologically distinct regions, where the upper-tail probabilities either resemble those of Gaussian or Poisson random variables, or are naturally described by the probability of appearance of a small set that contains nearly all of the excess $t$ progressions. We employ a variety of tools from probability theory, including classical tilting arguments and martingale concentration inequalities. However, the main technical innovation is a combinatorial result that establishes a stronger version of `entropic stability' 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"。
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
We consider Non Autonomous Conformal Iterative Function Systems (NACIFS) and their limit set. Our main concern is harmonic measure and its dimensions : Hausdorff and Packing. We prove that this two dimensions are continuous under perturbations and that they verify Bowen's and Manning's type formulas. In order to do so we prove general results about measures, and more generally about positive functionals, defined on a symbolic space, developing tools from thermodynamical formalism in a non-autonomous setting.
{"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":null,"pages":null},"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}
Consider a random walk in $mathbb{R}^d$ that starts at the origin and whose increment distribution assigns zero probability to any affine hyperplane. We solve Sylvester's problem for these random walks by showing that the probability that the first $d+2$ steps of the walk are in convex position is equal to $1-frac{2}{(d+1)!}$. The analogous result also holds for random bridges of length $d+2$, so long as the joint increment distribution is exchangeable.
{"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":null,"pages":null},"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}
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