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Asymptotic laws of summands I: square integrable independent random variables 求和的渐近律I:平方可积独立随机变量
arXiv: Probability Pub Date : 2021-08-22 DOI: 10.16929/hs/imhotep.2021.x.002
Aladji Babacar Niang, G. Lo, Moumouni Diallo
{"title":"Asymptotic laws of summands I: square integrable independent random variables","authors":"Aladji Babacar Niang, G. Lo, Moumouni Diallo","doi":"10.16929/hs/imhotep.2021.x.002","DOIUrl":"https://doi.org/10.16929/hs/imhotep.2021.x.002","url":null,"abstract":"This paper is part of series on self-contained papers in which a large part, if not the full extent, of the asymptotic limit theory of summands of independent random variables is exposed. Each paper of the series may be taken as review exposition but specially as a complete exposition expect a few exterior resources. For graduate students and for researchers (beginners or advanced), any paper of the series should be considered as a basis for constructing new results. The contents are taken from advanced books but the organization and the proofs use more recent tools, are given in more details and do not systematically follow previous one. Sometimes, theorems are completed and innovated","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81260652","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: 2
On cyclic and nontransitive probabilities 关于循环概率和非传递概率
arXiv: Probability Pub Date : 2020-12-09 DOI: 10.2140/involve.2021.14.327
P. Vuksanović, A. Hildebrand
{"title":"On cyclic and nontransitive probabilities","authors":"P. Vuksanović, A. Hildebrand","doi":"10.2140/involve.2021.14.327","DOIUrl":"https://doi.org/10.2140/involve.2021.14.327","url":null,"abstract":"Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $(x_1,dots,x_n) in[0,1]^n$ textit{cyclic} if there exist independent random variables $U_1,dots, U_n$ with $P(U_i=U_j)=0$ for $inot=j$ such that $P(U_{i+1}>U_i)=x_i$ for $i=1,dots,n-1$ and $P(U_1>U_n)=x_n$. We call the tuple $(x_1,dots,x_n)$ textit{nontransitive} if it is cyclic and in addition satisfies $x_i>1/2$ for all $i$. \u0000Let $p_n$ (resp.~$p_n^*$) denote the probability that a randomly chosen $n$-tuple $(x_1,dots,x_n)in[0,1]^n$ is cyclic (resp.~nontransitive). We determine $p_3$ and $p_3^*$ exactly, while for $nge4$ we give upper and lower bounds for $p_n$ that show that $p_n$ converges to $1$ as $ntoinfty$. We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83619656","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: 2
At the edge of a one-dimensional jellium 在一维凝胶的边缘
arXiv: Probability Pub Date : 2020-12-08 DOI: 10.3150/21-BEJ1397
Djalil CHAFAÏ, David Garc'ia-Zelada, Paul Jung
{"title":"At the edge of a one-dimensional jellium","authors":"Djalil CHAFAÏ, David Garc'ia-Zelada, Paul Jung","doi":"10.3150/21-BEJ1397","DOIUrl":"https://doi.org/10.3150/21-BEJ1397","url":null,"abstract":"We consider a one-dimensional classical Wigner jellium, not necessarily charge neutral, for which the electrons are allowed to exist beyond the support of the background charge. The model can be seen as a one-dimensional Coulomb gas in which the external field is generated by a smeared background on an interval. It is a true one-dimensional Coulomb gas and not a one-dimensional log-gas. We first observe that the system exists if and only if the total background charge is greater than the number of electrons minus one. Moreover we obtain a R'enyi-type probabilistic representation for the order statistics of the particle system beyond the support of the background. Furthermore, for various backgrounds, we show convergence to point processes, at the edge of the support of the background. In particular, this provides asymptotic analysis of the fluctuations of the right-most particle. Our analysis reveals that these fluctuations are not universal, in the sense that depending on the background, the tails range anywhere from exponential to Gaussian-like behavior, including for instance Tracy-Widom-like behavior.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79919122","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: 5
Population genetic models of dormancy 休眠的种群遗传模型
arXiv: Probability Pub Date : 2020-12-01 DOI: 10.4171/ecr/17-1/12
J. Blath, N. Kurt
{"title":"Population genetic models of dormancy","authors":"J. Blath, N. Kurt","doi":"10.4171/ecr/17-1/12","DOIUrl":"https://doi.org/10.4171/ecr/17-1/12","url":null,"abstract":"In the present article, we investigate the effects of dormancy on an abstract population genetic level. We first provide a short review of seed bank models in population genetics, and the role of dormancy for the interplay of evolutionary forces in general, before we discuss two recent paradigmatic models, referring to spontaneous resp. simultaneous switching of individuals between the active and the dormant state. We show that both mechanisms give rise to non-trivial mathematical objects, namely the (continuous) seed bank diffusion and the seed bank diffusion with jumps, as well as their dual processes, the seed bank coalescent and the seed bank coalescent with simultaneous switching.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76927202","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: 8
Optimal and algorithmic norm regularization of random matrices 随机矩阵的最优和算法范数正则化
arXiv: Probability Pub Date : 2020-11-30 DOI: 10.1090/proc/15964
Vishesh Jain, A. Sah, Mehtaab Sawhney
{"title":"Optimal and algorithmic norm regularization of random matrices","authors":"Vishesh Jain, A. Sah, Mehtaab Sawhney","doi":"10.1090/proc/15964","DOIUrl":"https://doi.org/10.1090/proc/15964","url":null,"abstract":"Let $A$ be an $ntimes n$ random matrix whose entries are i.i.d. with mean $0$ and variance $1$. We present a deterministic polynomial time algorithm which, with probability at least $1-2exp(-Omega(epsilon n))$ in the choice of $A$, finds an $epsilon n times epsilon n$ sub-matrix such that zeroing it out results in $widetilde{A}$ with [|widetilde{A}| = Oleft(sqrt{n/epsilon}right).] Our result is optimal up to a constant factor and improves previous results of Rebrova and Vershynin, and Rebrova. We also prove an analogous result for $A$ a symmetric $ntimes n$ random matrix whose upper-diagonal entries are i.i.d. with mean $0$ and variance $1$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74062573","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
From Ball's cube slicing inequality to Khinchin-type inequalities for negative moments 从Ball的立方体切片不等式到负矩的khinchin型不等式
arXiv: Probability Pub Date : 2020-11-24 DOI: 10.1016/J.JFA.2021.109185
Giorgos Chasapis, Hermann Konig, T. Tkocz
{"title":"From Ball's cube slicing inequality to Khinchin-type inequalities for negative moments","authors":"Giorgos Chasapis, Hermann Konig, T. Tkocz","doi":"10.1016/J.JFA.2021.109185","DOIUrl":"https://doi.org/10.1016/J.JFA.2021.109185","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74067500","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: 15
The speed of random walk on Galton-Watson trees with vanishing conductances 具有消失电导的高尔顿-沃森树的随机漫步速度
arXiv: Probability Pub Date : 2020-11-20 DOI: 10.1214/21-ejp645
Tabea Glatzel, J. Nagel
{"title":"The speed of random walk on Galton-Watson trees with vanishing conductances","authors":"Tabea Glatzel, J. Nagel","doi":"10.1214/21-ejp645","DOIUrl":"https://doi.org/10.1214/21-ejp645","url":null,"abstract":"In this paper we consider random walks on Galton-Watson trees with random conductances. On these trees, the distance of the walker to the root satisfies a law of large numbers with limit the effective velocity, or speed of the walk. We study the regularity of the speed as a function of the distribution of conductances, in particular when the distribution of conductances converges to a non-elliptic limit.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85050221","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
Poincaré inequalities and normal approximation for weighted sums 加权和的庞加莱不等式和正态逼近
arXiv: Probability Pub Date : 2020-11-18 DOI: 10.1214/20-ejp549
S. Bobkov, G. Chistyakov, Friedrich Götze
{"title":"Poincaré inequalities and normal approximation for weighted sums","authors":"S. Bobkov, G. Chistyakov, Friedrich Götze","doi":"10.1214/20-ejp549","DOIUrl":"https://doi.org/10.1214/20-ejp549","url":null,"abstract":"Under Poincare-type conditions, upper bounds are explored for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. Based on improved concentration inequalities on high-dimensional Euclidean spheres, the results extend and refine previous results to non-symmetric models.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72906642","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: 6
Boundary behaviors for a class of continuous-state nonlinear branching processes in critical cases 一类临界情况下连续状态非线性分支过程的边界行为
arXiv: Probability Pub Date : 2020-11-12 DOI: 10.1214/21-ECP374
Shaojuan Ma, Xu Yang, Xiaowen Zhou
{"title":"Boundary behaviors for a class of continuous-state nonlinear branching processes in critical cases","authors":"Shaojuan Ma, Xu Yang, Xiaowen Zhou","doi":"10.1214/21-ECP374","DOIUrl":"https://doi.org/10.1214/21-ECP374","url":null,"abstract":"Using Foster-Lyapunov techniques we establish new conditions on non-extinction, non-explosion, coming down from infinity and staying infinite, respectively, for the general continuous-state nonlinear branching processes introduced in Li et al. (2019). These results can be applied to identify boundary behaviors for the critical cases of the above nonlinear branching processes with power rate functions driven by Brownian motion and (or) stable Poisson random measure, which was left open in Li et al. (2019). In particular, we show that even in the critical cases, a phase transition happens between coming down from infinity and staying infinite.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76659588","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: 3
A general stochastic matching model on multigraphs 多图上的一般随机匹配模型
arXiv: Probability Pub Date : 2020-11-10 DOI: 10.30757/alea.v18-49
Jocelyn Begeot, Irène Marcovici, P. Moyal, Youssef Rahme
{"title":"A general stochastic matching model on multigraphs","authors":"Jocelyn Begeot, Irène Marcovici, P. Moyal, Youssef Rahme","doi":"10.30757/alea.v18-49","DOIUrl":"https://doi.org/10.30757/alea.v18-49","url":null,"abstract":"We extend the general stochastic matching model on graphs introduced in (Mairesse and Moyal, 2016), to matching models on multigraphs, that is, graphs with self-loops. \u0000The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. \u0000Necessary and sufficient stability conditions are provided, together with the explicit form of the stationary probability in the case where the matching policy is `First Come, First Matched'.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86798080","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: 7
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