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Gaussian Asymptotics of Jack Measures on Partitions from Weighted Enumeration of Ribbon Paths 带状路径加权枚举分区上Jack测度的高斯渐近性
arXiv: Probability Pub Date : 2020-10-26 DOI: 10.1093/IMRN/RNAB300
Alexander Moll
{"title":"Gaussian Asymptotics of Jack Measures on Partitions from Weighted Enumeration of Ribbon Paths","authors":"Alexander Moll","doi":"10.1093/IMRN/RNAB300","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB300","url":null,"abstract":"In this paper we determine two asymptotic results for Jack measures on partitions, a model defined by two specializations of Jack polynomials proposed by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming these two specializations are the same, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to diverging, fixed, and vanishing values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call \"ribbon paths\", show for general Jack measures that certain joint cumulants are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov-Sklyanin's spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack-Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy-Dolk{e}ga.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83954424","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
A dual Yamada–Watanabe theorem for Lévy driven stochastic differential equations 随机微分方程的对偶Yamada-Watanabe定理
arXiv: Probability Pub Date : 2020-10-22 DOI: 10.1214/21-ECP384
David Criens
{"title":"A dual Yamada–Watanabe theorem for Lévy driven stochastic differential equations","authors":"David Criens","doi":"10.1214/21-ECP384","DOIUrl":"https://doi.org/10.1214/21-ECP384","url":null,"abstract":"We prove a dual Yamada--Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Levy processes.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79552349","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
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":null,"pages":null},"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":null,"pages":null},"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
On nonnegative solutions of SDDEs with an application to CARMA processes SDDEs的非负解及其在CARMA过程中的应用
arXiv: Probability Pub Date : 2020-10-16 DOI: 10.15559/21-VMSTA177
M. S. Nielsen, V. Rohde
{"title":"On nonnegative solutions of SDDEs with an application to CARMA processes","authors":"M. S. Nielsen, V. Rohde","doi":"10.15559/21-VMSTA177","DOIUrl":"https://doi.org/10.15559/21-VMSTA177","url":null,"abstract":"This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are non-negative. While, to the best of our knowledge, no simple non-negativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $pgeq 3$. Finally, we extend the result to a multidimensional setting.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79888932","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
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
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