Pub Date : 2021-08-22DOI: 10.16929/hs/imhotep.2021.x.002
Aladji Babacar Niang, G. Lo, Moumouni Diallo
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
{"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":"8 1","pages":""},"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}
Pub Date : 2020-12-09DOI: 10.2140/involve.2021.14.327
P. Vuksanović, A. Hildebrand
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 $i$. Let $p_n$ (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.
{"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":"1 1","pages":""},"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}
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.
{"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":"1 1","pages":""},"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}
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.
{"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":"3 1","pages":""},"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}
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 $1$.
设$A$为一个$ntimes n$随机矩阵，其项为i.i.d.，均值为$0$，方差为$1$。我们提出了一种确定性多项式时间算法，该算法在$A$的选择中至少以$1-2exp(-Omega(epsilon n))$的概率找到$epsilon n times epsilon n$子矩阵，使得在$widetilde{A}$中使用[|widetilde{A}| = Oleft(sqrt{n/epsilon}right).]将其归零。我们的结果是最优的，直到一个常数因子，并改进了以前的Rebrova和Vershynin以及Rebrova的结果。对于一个对称的$ntimes n$随机矩阵$A$，我们也证明了一个类似的结果，该矩阵的上对角线项为i.i.d，均值为$0$，方差为$1$。
{"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":"28 1","pages":""},"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}
Pub Date : 2020-11-24DOI: 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":"72 1","pages":""},"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}
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.
{"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":"34 1","pages":""},"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}
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.
{"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":"287 1","pages":""},"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}
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.
使用Foster-Lyapunov技术，我们分别为Li et al.(2019)中引入的一般连续状态非线性分支过程建立了非消灭、非爆炸、从无穷下降和保持无穷的新条件。这些结果可用于识别上述由布朗运动和(或)稳定泊松随机测度驱动的功率速率函数的非线性分支过程的临界情况的边界行为，Li et al.(2019)未对此进行开放。特别地，我们证明了即使在临界情况下，相变也发生在从无穷向下和保持无穷之间。
{"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":"95 1","pages":""},"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}
Jocelyn Begeot, Irène Marcovici, P. Moyal, Youssef Rahme
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. The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. Necessary 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'.
{"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":"3 1","pages":""},"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}
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