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Real roots near the unit circle of random polynomials 随机多项式单位圆附近的实根
arXiv: Probability Pub Date : 2020-10-21 DOI: 10.1090/TRAN/8379
Marcus Michelen
{"title":"Real roots near the unit circle of random polynomials","authors":"Marcus Michelen","doi":"10.1090/TRAN/8379","DOIUrl":"https://doi.org/10.1090/TRAN/8379","url":null,"abstract":"Let $f_n(z) = sum_{k = 0}^n varepsilon_k z^k$ be a random polynomial where $varepsilon_0,ldots,varepsilon_n$ are i.i.d. random variables with $mathbb{E} varepsilon_1 = 0$ and $mathbb{E} varepsilon_1^2 = 1$. Letting $r_1, r_2,ldots, r_k$ denote the real roots of $f_n$, we show that the point process defined by ${|r_1| - 1,ldots, |r_k| - 1 }$ converges to a non-Poissonian limit on the scale of $n^{-1}$ as $n to infty$. Further, we show that for each $delta > 0$, $f_n$ has a real root within $Theta_{delta}(1/n)$ of the unit circle with probability at least $1 - delta$. This resolves a conjecture of Shepp and Vanderbei from 1995 by confirming its weakest form and refuting its strongest form.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"55 33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88492665","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
Note on local mixing techniques for stochastic differential equations 随机微分方程的局部混合技术
arXiv: Probability Pub Date : 2020-10-19 DOI: 10.15559/21-VMSTA174
A. Veretennikov
{"title":"Note on local mixing techniques for stochastic differential equations","authors":"A. Veretennikov","doi":"10.15559/21-VMSTA174","DOIUrl":"https://doi.org/10.15559/21-VMSTA174","url":null,"abstract":"This paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). They all work in dimension $dge 1$, although, in $d=1$ the most natural way is to use intersections of trajectories, which requires nothing but strong Markov property and non-degeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,ldots$, or by arranging special stopping time sequences and to use local Markov -- Dobrushin's (MD) condition. Further applications may be based on one or another version of the MD condition. For studies of convergence and mixing rates the (Markov) process must be strong Markov and recurrent; however, recurrence is a separate issue which is not discussed in this paper.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74567529","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: 4
Realized cumulants for martingales 鞅的已实现累积量
arXiv: Probability Pub Date : 2020-10-16 DOI: 10.1214/21-ECP382
M. Fukasawa, Kazuki Matsushita
{"title":"Realized cumulants for martingales","authors":"M. Fukasawa, Kazuki Matsushita","doi":"10.1214/21-ECP382","DOIUrl":"https://doi.org/10.1214/21-ECP382","url":null,"abstract":"Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"83 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89850550","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
Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains 连续时间马尔可夫链曲率维条件下的熵信息不等式
arXiv: Probability Pub Date : 2020-10-13 DOI: 10.1214/21-EJP627
Frederic Weber
{"title":"Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains","authors":"Frederic Weber","doi":"10.1214/21-EJP627","DOIUrl":"https://doi.org/10.1214/21-EJP627","url":null,"abstract":"In the setting of reversible continuous-time Markov chains, the $CD_Upsilon$ condition has been shown recently to be a consistent analogue to the Bakry-Emery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the $CD_Upsilon(kappa,F)$ condition, where the dimension term is expressed by a so called $CD$-function $F$. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84425934","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
Ergodicity and Polynomial Convergence Rate of Generalized Markov Modulated Poisson Processes 广义马尔可夫调制泊松过程的遍历性和多项式收敛速度
arXiv: Probability Pub Date : 2020-10-12 DOI: 10.1007/978-3-030-66242-4_29
G. Zverkina
{"title":"Ergodicity and Polynomial Convergence Rate of Generalized Markov Modulated Poisson Processes","authors":"G. Zverkina","doi":"10.1007/978-3-030-66242-4_29","DOIUrl":"https://doi.org/10.1007/978-3-030-66242-4_29","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"378 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84950128","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
Spatial averages for the parabolic Anderson model driven by rough noise 粗糙噪声驱动下抛物型Anderson模型的空间平均
arXiv: Probability Pub Date : 2020-10-12 DOI: 10.30757/ALEA.V18-33
D. Nualart, Xiaoming Song, Guangqu Zheng
{"title":"Spatial averages for the parabolic Anderson model driven by rough noise","authors":"D. Nualart, Xiaoming Song, Guangqu Zheng","doi":"10.30757/ALEA.V18-33","DOIUrl":"https://doi.org/10.30757/ALEA.V18-33","url":null,"abstract":"In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0 in (1/2,1)$, $H_1in (0,1/2)$ and $H_0 + H_1 > 3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83302889","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: 11
Weakly reinforced Pólya urns on countable networks 弱增强Pólya在可数网络上运行
arXiv: Probability Pub Date : 2020-10-07 DOI: 10.1214/21-ECP404
Yannick Couzini'e, C. Hirsch
{"title":"Weakly reinforced Pólya urns on countable networks","authors":"Yannick Couzini'e, C. Hirsch","doi":"10.1214/21-ECP404","DOIUrl":"https://doi.org/10.1214/21-ECP404","url":null,"abstract":"We study the long-time asymptotics of a network of weakly reinforced Polya urns. In this system, which extends the WARM introduced by R. van der Hofstad et. al. (2016) to countable networks, the nodes fire at times given by a Poisson point process. When a node fires, one of the incident edges is selected with a probability proportional to its weight raised to a power $alpha < 1$, and then this weight is increased by $1$. \u0000We show that for $alpha < 1/2$ on a network of bounded degrees, every edge is reinforced a positive proportion of time, and that the limiting proportion can be interpreted as an equilibrium in a countable network. Moreover, in the special case of regular graphs, this homogenization remains valid beyond the threshold $alpha = 1/2$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87646268","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: 4
A study of backward stochastic differential equation on a Riemannian manifold 黎曼流形上倒向随机微分方程的研究
arXiv: Probability Pub Date : 2020-10-06 DOI: 10.1214/21-ejp649
Xin Chen, Wenjie Ye
{"title":"A study of backward stochastic differential equation on a Riemannian manifold","authors":"Xin Chen, Wenjie Ye","doi":"10.1214/21-ejp649","DOIUrl":"https://doi.org/10.1214/21-ejp649","url":null,"abstract":"Suppose $N$ is a compact Riemannian manifold, in this paper we will introduce the definition of $N$-valued BSDE and $L^2(mathbb{T}^m;N)$-valued BSDE for which the solution are not necessarily staying in only one local coordinate. Moreover, the global existence of a solution to $L^2(mathbb{T}^m;N)$-valued BSDE will be proved without any convexity condition on $N$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"132 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89124467","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: 4
Upper bounds for the maximum deviation of the Pearcey process 皮尔斯过程最大偏差的上界
arXiv: Probability Pub Date : 2020-09-28 DOI: 10.1142/s2010326321500398
C. Charlier
{"title":"Upper bounds for the maximum deviation of the Pearcey process","authors":"C. Charlier","doi":"10.1142/s2010326321500398","DOIUrl":"https://doi.org/10.1142/s2010326321500398","url":null,"abstract":"The Pearcey process is a universal point process in random matrix theory and depends on a parameter $rho in mathbb{R}$. Let $N(x)$ be the random variable that counts the number of points in this process that fall in the interval $[-x,x]$. In this note, we establish the following global rigidity upper bound: begin{align*} lim_{s to infty}mathbb Pleft(sup_{x> s}left|frac{N(x)-big( frac{3sqrt{3}}{4pi}x^{frac{4}{3}}-frac{sqrt{3}rho}{2pi}x^{frac{2}{3}} big)}{log x}right| leq frac{4sqrt{2}}{3pi} + epsilon right) = 1, end{align*} where $epsilon > 0$ is arbitrary. We also obtain a similar upper bound for the maximum deviation of the points, and a central limit theorem for the individual fluctuations. The proof is short and combines a recent result of Dai, Xu and Zhang with another result of Charlier and Claeys.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"14 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91478968","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
Extremal Indices in the Series Scheme and their Applications 级数格式中的极值索引及其应用
arXiv: Probability Pub Date : 2020-09-20 DOI: 10.14357/19922264150305
A. Lebedev
{"title":"Extremal Indices in the Series Scheme and their Applications","authors":"A. Lebedev","doi":"10.14357/19922264150305","DOIUrl":"https://doi.org/10.14357/19922264150305","url":null,"abstract":"We generalize the concept of extremal index of a stationary random sequence to the series scheme of identically distributed random variables with random series sizes tending to infinity in probability. We introduce new extremal indices through two definitions generalizing the basic properties of the classical extremal index. We prove some useful properties of the new extremal indices. We show how the behavior of aggregate activity maxima on random graphs (in information network models) and the behavior of maxima of random particle scores in branching processes (in biological population models) can be described in terms of the new extremal indices. We also obtain new results on models with copulas and threshold models. We show that the new indices can take different values for the same system, as well as values greater than one.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86713835","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
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