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.
本文确定了由Borodin-Olshanski在[European J. Combin. 26.6(2005): 795-834]中提出的Jack多项式的两个专门化所定义的分区上的Jack测度的两个渐近结果。假设这两种专门化是相同的，我们在与Jack参数的发散值、固定值和消失值相关的三个渐近区域中推导出这些随机分区的各向异性剖面的极限形状和高斯波动。为此，我们引入了莫兹金路径的一种推广，我们称之为“带状路径”，表明对于一般的Jack测度，某些联合累积量是$n$位置上具有$n-1+g$配对的连接带状路径的加权和，并分别从$(n,g)=(1,0)$和$(2,0)$的贡献中得出我们的两个结果。我们的分析使用了纳扎罗夫-斯克里亚宁的杰克多项式谱理论。因此，我们对Schur测度、Plancherel测度和Jack-Plancherel测度的几个结果给出了新的证明。此外，我们将带状路径的加权和与最近由chapy - dol k{e}ga引入的非定向真实曲面上映射的带状图的加权和联系起来。
{"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":"72 1","pages":""},"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}
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.
{"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":"47 1","pages":""},"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}
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.
{"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}
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.
{"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}
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.
{"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":"9 1","pages":""},"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}
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.
{"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}
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.
{"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}
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.
{"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}
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 $1$. We 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 $alpha = 1/2$.
研究了一类弱增强Polya瓮网络的长期渐近性。在该系统中，将R. van der Hofstad等人(2016)引入的WARM扩展到可计数网络，节点在泊松点过程给定的时间内触发。当一个节点触发时，其中一个事件边被选择，其概率与它的权重成正比，提高到幂$alpha < 1$，然后这个权重增加$1$。我们证明了在有界度网络上，对于$alpha < 1/2$，每条边都被强化了正比例的时间，并且这个极限比例可以解释为可数网络中的平衡。此外，在正则图的特殊情况下，这种均质化在阈值$alpha = 1/2$之外仍然有效。
{"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}
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