{"title":"On the explosion of the number of fragments in simple exchangeable fragmentation-coagulation processes","authors":"Clément Foucart, Xiaowen Zhou","doi":"10.1214/21-aihp1191","DOIUrl":"https://doi.org/10.1214/21-aihp1191","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"12 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89456206","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 define intermediate SLEκ(ρ) and reversed intermediate SLEκ(ρ) processes using Appell-Lauricella multiple hypergeometric functions, and use them to describe the timereversal of multiple-force-point chordal SLEκ(ρ) curves in the case that all force points are on the boundary and lie on the same side of the initial point, and κ and ρ = (ρ1, . . . , ρm) satisfy that either κ ∈ (0, 4] and kj=1 ρj > −2 for all 1 ≤ k ≤ m, or κ ∈ (4, 8) and ∑k j=1 ρj ≥ κ2 − 2 for all 1 ≤ k ≤ m.
{"title":"Time-reversal of multiple-force-point SLEκ(ρ_) with all force points lying on the same side","authors":"Dapeng Zhan","doi":"10.1214/21-aihp1170","DOIUrl":"https://doi.org/10.1214/21-aihp1170","url":null,"abstract":"We define intermediate SLEκ(ρ) and reversed intermediate SLEκ(ρ) processes using Appell-Lauricella multiple hypergeometric functions, and use them to describe the timereversal of multiple-force-point chordal SLEκ(ρ) curves in the case that all force points are on the boundary and lie on the same side of the initial point, and κ and ρ = (ρ1, . . . , ρm) satisfy that either κ ∈ (0, 4] and kj=1 ρj > −2 for all 1 ≤ k ≤ m, or κ ∈ (4, 8) and ∑k j=1 ρj ≥ κ2 − 2 for all 1 ≤ k ≤ m.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"48 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78247321","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}
Testing uniformity on the unit sphere of R is a fundamental problem in directional statistics. In the framework of axial data, the most classical test of uniformity is the Bingham [8] test. Remarkably, this test does not need any modification to meet asymptotically the target null size in high-dimensional scenarios where p = pn diverges to infinity with the sample size n. However, while the non-null asymptotic behaviour of the Bingham test is well understood in standard asymptotic scenarios where n diverges to infinity with p fixed, nothing is known on the power of this test in high dimensions, not even under standard parametric alternatives such as Watson distributions. In this work, we therefore study the non-null behaviour of the Bingham test in high dimensions. First, we consider a semiparametric class of alternatives that includes Watson alternatives and we derive a local asymptotic normality (LAN) property. An application of Le Cam’s third lemma reveals that the Bingham test is blind to the corresponding contiguous alternatives, though. By using martingale central limit theorems, we therefore study the non-null behaviour of the Bingham test under more severe alternatives. Far from restricting to the aforementioned semiparametric alternatives, our results cover a broad class of rotationally symmetric alternatives, which allows us to consider non-axial alternatives, too. In every distributional framework we consider, the “detection threshold” of the Bingham test is identified and a comparison with the classical test of uniformity for non-axial data, namely the Rayleigh [40] test, is made possible. In the framework of axial data, we derive a lower bound on the minimax separation rate and establish that the Bingham test is minimax rate-optimal in the class of Watson distributions. MSC 2010 subject classifications: Primary 62H11, 62F05; secondary 62E20.
{"title":"Testing uniformity on high-dimensional spheres: The non-null behaviour of the Bingham test","authors":"C. Cutting, D. Paindaveine, Thomas Verdebout","doi":"10.1214/21-aihp1168","DOIUrl":"https://doi.org/10.1214/21-aihp1168","url":null,"abstract":"Testing uniformity on the unit sphere of R is a fundamental problem in directional statistics. In the framework of axial data, the most classical test of uniformity is the Bingham [8] test. Remarkably, this test does not need any modification to meet asymptotically the target null size in high-dimensional scenarios where p = pn diverges to infinity with the sample size n. However, while the non-null asymptotic behaviour of the Bingham test is well understood in standard asymptotic scenarios where n diverges to infinity with p fixed, nothing is known on the power of this test in high dimensions, not even under standard parametric alternatives such as Watson distributions. In this work, we therefore study the non-null behaviour of the Bingham test in high dimensions. First, we consider a semiparametric class of alternatives that includes Watson alternatives and we derive a local asymptotic normality (LAN) property. An application of Le Cam’s third lemma reveals that the Bingham test is blind to the corresponding contiguous alternatives, though. By using martingale central limit theorems, we therefore study the non-null behaviour of the Bingham test under more severe alternatives. Far from restricting to the aforementioned semiparametric alternatives, our results cover a broad class of rotationally symmetric alternatives, which allows us to consider non-axial alternatives, too. In every distributional framework we consider, the “detection threshold” of the Bingham test is identified and a comparison with the classical test of uniformity for non-axial data, namely the Rayleigh [40] test, is made possible. In the framework of axial data, we derive a lower bound on the minimax separation rate and establish that the Bingham test is minimax rate-optimal in the class of Watson distributions. MSC 2010 subject classifications: Primary 62H11, 62F05; secondary 62E20.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"97 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85750048","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 well-posedness of a nonlinear one dimensional stochastic heat equation driven by Gaussian noise: ∂u ∂t = ∂ u ∂x2 + σ(u)Ẇ , where Ẇ is white in time and fractional in space with Hurst parameter H ∈ ( 1 4 , 1 2 ). In a recent paper [12] by Hu, Huang, Lê, Nualart and Tindel a technical and unusual condition of σ(0) = 0 was assumed which is critical in their approach. The main effort of this paper is to remove this condition. The idea is to work on a weighted space Z λ,T for some power decay weight λ(x) = cH(1 + |x| 2)H−1. In addition, when σ(u) = 1 we obtain the exact asympotics of the solution uadd(t, x) as t and x go to infinity. In particular, we find the exact growth of sup|x|≤L |uadd(t, x)| and the sharp growth rate for the Hölder coefficients, namely, sup|x|≤L |uadd(t,x+h)−uadd(t,x)| |h|β and sup|x|≤L |uadd(t+τ,x)−uadd(t,x)| τα . Abstract. Nous étudions une équation de chaleur stochastique á une dimension spatiale non linéaire entrânée par le bruit gaussien: ∂u ∂t = ∂ u ∂x2 + σ(u)Ẇ , où Ẇ est blanc dans le temps et fractionnaire dans le espace avec le paramètre Hurst H ∈ ( 1 4 , 1 2 ). Dans un article récent [12] par Hu, Huang, Lê, Nualart et Tindel une condition technique et inhabituelle de σ(0) = 0 a été supposé, ce qui est critique dans leur approche. Le principal effort de ce document est de supprimer cette condition. L’idée est de travailler sur un espace pondéré Z λ,T pour un certain poids de décroissance de puissance λ(x) = cH(1+|x|). Lorsque σ(u) = 1 nous obtenons les asympotiques exacts de la solution uadd(t, x) as t et x vont l’infini. En particulier, nous trouvons la croissance exacte de sup|x|≤L |uadd(t, x)| et la croissance exacte des coefficients de Hölder, c’est-àdire, sup|x|≤L |uadd(t,x+h)−uadd(t,x)| |h|β et sup|x|≤L |uadd(t+τ,x)−uadd(t,x)| τα . Nous étudions une équation de chaleur stochastique á une dimension spatiale non linéaire entrânée par le bruit gaussien: ∂u ∂t = ∂ u ∂x2 + σ(u)Ẇ , où Ẇ est blanc dans le temps et fractionnaire dans le espace avec le paramètre Hurst H ∈ ( 1 4 , 1 2 ). Dans un article récent [12] par Hu, Huang, Lê, Nualart et Tindel une condition technique et inhabituelle de σ(0) = 0 a été supposé, ce qui est critique dans leur approche. Le principal effort de ce document est de supprimer cette condition. L’idée est de travailler sur un espace pondéré Z λ,T pour un certain poids de décroissance de puissance λ(x) = cH(1+|x|). Lorsque σ(u) = 1 nous obtenons les asympotiques exacts de la solution uadd(t, x) as t et x vont l’infini. En particulier, nous trouvons la croissance exacte de sup|x|≤L |uadd(t, x)| et la croissance exacte des coefficients de Hölder, c’est-àdire, sup|x|≤L |uadd(t,x+h)−uadd(t,x)| |h|β et sup|x|≤L |uadd(t+τ,x)−uadd(t,x)| τα .
我们研究了一个由高斯噪声驱动的非线性一维随机热方程的适定性:∂u∂t =∂u∂x2 + σ(u)Ẇ，其中Ẇ在时间上是白色的，在空间上是分数阶的，Hurst参数H∈(1,4,1,2)。在Hu, Huang, Lê， Nualart和Tindel最近的一篇论文[12]中，假设σ(0) = 0的技术和异常条件，这是他们方法的关键。本文的主要工作就是消除这种状况。这个想法是在一个加权空间Z λ T上工作，对于一些功率衰减权λ(x) = cH(1 + |x| 2)H−1。另外，当σ(u) = 1时，我们得到了解uadd(t, x)在t和x趋于无穷时的确切渐近性。特别地，我们发现sup|x|≤L |uadd(t,x)|的精确增长和Hölder系数的急剧增长，即sup|x|≤L |uadd(t,x+h) - uadd(t,x)| |h|β和sup|x|≤L |uadd(t+τ，x) - uadd(t,x)| τα。摘要Nous日新月异的空间非空间型的与其他所有的空间型的，与其他所有的空间型的相同:∂u∂t =∂u∂x2 + σ(u)Ẇ， où Ẇ est blanc dans le temps et partitionnaire dans le espace avec le param tre Hurst H∈(1,1,12)。[12]胡佩尔，黄，Lê， Nualart等。一种条件技术et inhabituelle de σ(0) = 0 a - samuest est方法。主要工作是编制文件，测试供应商的测试条件。L ' idsamuest de travailler sur un espace pondsamuise r Z λ，T pour on certain poids de dsamuise de puissance λ(x) = cH(1+|x|)。洛斯克σ(u) = 1，它的渐近性与解(t, x)的渐近性一致，因为t = x =∞。具体来说，nous trouvons la croissance exacte de sup|x|≤L |uadd(t,x)| et la croissance exacte des coefficients de Hölder, c 'est -àdire, sup|x|≤L |uadd(t,x+h) - uadd(t,x)| |h|β et sup|x|≤L |uadd(t+τ，x) - uadd(t,x)| τα。Nous日新月异的空间非空间型的与其他所有的空间型的，与其他所有的空间型的相同:∂u∂t =∂u∂x2 + σ(u)Ẇ， où Ẇ est blanc dans le temps et partitionnaire dans le espace avec le param tre Hurst H∈(1,1,12)。[12]胡佩尔，黄，Lê， Nualart等。一种条件技术et inhabituelle de σ(0) = 0 a - samuest est方法。主要工作是编制文件，测试供应商的测试条件。L ' idsamuest de travailler sur un espace pondsamuise r Z λ，T pour on certain poids de dsamuise de puissance λ(x) = cH(1+|x|)。洛斯克σ(u) = 1，它的渐近性与解(t, x)的渐近性一致，因为t = x =∞。具体来说，nous trouvons la croissance exacte de sup|x|≤L |uadd(t,x)| et la croissance exacte des coefficients de Hölder, c 'est -àdire, sup|x|≤L |uadd(t,x+h) - uadd(t,x)| |h|β et sup|x|≤L |uadd(t+τ，x) - uadd(t,x)| τα。
{"title":"Stochastic heat equation with general rough noise","authors":"Yaozhong Hu, Xiongrui Wang","doi":"10.1214/21-aihp1161","DOIUrl":"https://doi.org/10.1214/21-aihp1161","url":null,"abstract":"We study the well-posedness of a nonlinear one dimensional stochastic heat equation driven by Gaussian noise: ∂u ∂t = ∂ u ∂x2 + σ(u)Ẇ , where Ẇ is white in time and fractional in space with Hurst parameter H ∈ ( 1 4 , 1 2 ). In a recent paper [12] by Hu, Huang, Lê, Nualart and Tindel a technical and unusual condition of σ(0) = 0 was assumed which is critical in their approach. The main effort of this paper is to remove this condition. The idea is to work on a weighted space Z λ,T for some power decay weight λ(x) = cH(1 + |x| 2)H−1. In addition, when σ(u) = 1 we obtain the exact asympotics of the solution uadd(t, x) as t and x go to infinity. In particular, we find the exact growth of sup|x|≤L |uadd(t, x)| and the sharp growth rate for the Hölder coefficients, namely, sup|x|≤L |uadd(t,x+h)−uadd(t,x)| |h|β and sup|x|≤L |uadd(t+τ,x)−uadd(t,x)| τα . Abstract. Nous étudions une équation de chaleur stochastique á une dimension spatiale non linéaire entrânée par le bruit gaussien: ∂u ∂t = ∂ u ∂x2 + σ(u)Ẇ , où Ẇ est blanc dans le temps et fractionnaire dans le espace avec le paramètre Hurst H ∈ ( 1 4 , 1 2 ). Dans un article récent [12] par Hu, Huang, Lê, Nualart et Tindel une condition technique et inhabituelle de σ(0) = 0 a été supposé, ce qui est critique dans leur approche. Le principal effort de ce document est de supprimer cette condition. L’idée est de travailler sur un espace pondéré Z λ,T pour un certain poids de décroissance de puissance λ(x) = cH(1+|x|). Lorsque σ(u) = 1 nous obtenons les asympotiques exacts de la solution uadd(t, x) as t et x vont l’infini. En particulier, nous trouvons la croissance exacte de sup|x|≤L |uadd(t, x)| et la croissance exacte des coefficients de Hölder, c’est-àdire, sup|x|≤L |uadd(t,x+h)−uadd(t,x)| |h|β et sup|x|≤L |uadd(t+τ,x)−uadd(t,x)| τα . Nous étudions une équation de chaleur stochastique á une dimension spatiale non linéaire entrânée par le bruit gaussien: ∂u ∂t = ∂ u ∂x2 + σ(u)Ẇ , où Ẇ est blanc dans le temps et fractionnaire dans le espace avec le paramètre Hurst H ∈ ( 1 4 , 1 2 ). Dans un article récent [12] par Hu, Huang, Lê, Nualart et Tindel une condition technique et inhabituelle de σ(0) = 0 a été supposé, ce qui est critique dans leur approche. Le principal effort de ce document est de supprimer cette condition. L’idée est de travailler sur un espace pondéré Z λ,T pour un certain poids de décroissance de puissance λ(x) = cH(1+|x|). Lorsque σ(u) = 1 nous obtenons les asympotiques exacts de la solution uadd(t, x) as t et x vont l’infini. En particulier, nous trouvons la croissance exacte de sup|x|≤L |uadd(t, x)| et la croissance exacte des coefficients de Hölder, c’est-àdire, sup|x|≤L |uadd(t,x+h)−uadd(t,x)| |h|β et sup|x|≤L |uadd(t+τ,x)−uadd(t,x)| τα .","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"11 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79939420","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 manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive non-asymptotic estimates for the corresponding time average $p$-th moments. The proof relies on coupling techniques such as coin-flip coupling, synchronous coupling and the Koml'os--Major--Tusn'ady coupling.
{"title":"Quantitative control of Wasserstein distance between Brownian motion and the Goldstein–Kac telegraph process","authors":"G. Barrera, J. Lukkarinen","doi":"10.1214/22-AIHP1288","DOIUrl":"https://doi.org/10.1214/22-AIHP1288","url":null,"abstract":"In this manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive non-asymptotic estimates for the corresponding time average $p$-th moments. The proof relies on coupling techniques such as coin-flip coupling, synchronous coupling and the Koml'os--Major--Tusn'ady coupling.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"11 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73944512","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 show that external randomization may enforce the convergence of test statistics to their limiting distributions in particular cases. This results in a sharper inference. Our approach is based on a central limit theorem for weighted sums. We apply our method to a family of rank-based test statistics and a family of phi-divergence test statistics and prove that, with overwhelming probability with respect to the external randomization, the randomized statistics converge at the rate $O(1/n)$ (up to some logarithmic factors) to the limiting chi-square distribution in Kolmogorov metric.
{"title":"Inference via randomized test statistics","authors":"Nikita Puchkin, V. Ulyanov","doi":"10.1214/22-aihp1299","DOIUrl":"https://doi.org/10.1214/22-aihp1299","url":null,"abstract":"We show that external randomization may enforce the convergence of test statistics to their limiting distributions in particular cases. This results in a sharper inference. Our approach is based on a central limit theorem for weighted sums. We apply our method to a family of rank-based test statistics and a family of phi-divergence test statistics and prove that, with overwhelming probability with respect to the external randomization, the randomized statistics converge at the rate $O(1/n)$ (up to some logarithmic factors) to the limiting chi-square distribution in Kolmogorov metric.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"46 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79680211","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}
The Quantum Symmetric Simple Exclusion Process (QSSEP) is a model of quantum particles hopping on a finite interval and satisfying the exclusion principle. Recently Bernard and Jin have studied the fluctuations of the invariant measure for this process, when the number of sites goes to infinity. These fluctuations are encoded into polynomials, for which they have given equations and proved that these equations determine the polynomials completely. In this paper, I give an explicit combinatorial formula for these polynomials, in terms of Schr"oder trees. I also show that, quite surprisingly, these polynomials can be interpreted as free cumulants of a family of commuting random variables.
{"title":"Combinatorics of the quantum symmetric simple exclusion process, associahedra and free cumulants","authors":"P. Biane","doi":"10.4171/aihpd/175","DOIUrl":"https://doi.org/10.4171/aihpd/175","url":null,"abstract":"The Quantum Symmetric Simple Exclusion Process (QSSEP) is a model of quantum particles hopping on a finite interval and satisfying the exclusion principle. Recently Bernard and Jin have studied the fluctuations of the invariant measure for this process, when the number of sites goes to infinity. These fluctuations are encoded into polynomials, for which they have given equations and proved that these equations determine the polynomials completely. In this paper, I give an explicit combinatorial formula for these polynomials, in terms of Schr\"oder trees. I also show that, quite surprisingly, these polynomials can be interpreted as free cumulants of a family of commuting random variables.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"53 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74961189","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 fine properties of the convergence of a high intensity shot noise field towards the Gaussian field with the same covariance structure. In particular we (i) establish a strong invariance principle, i.e. a quantitative coupling between a high intensity shot noise field and the Gaussian limit such that they are uniformly close on large domains with high probability, and (ii) use this to derive an asymptotic expansion for the critical level above which the excursion sets of the shot noise field percolate.
{"title":"Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields","authors":"R. Lachièze-Rey, S. Muirhead","doi":"10.1214/22-aihp1303","DOIUrl":"https://doi.org/10.1214/22-aihp1303","url":null,"abstract":"We study fine properties of the convergence of a high intensity shot noise field towards the Gaussian field with the same covariance structure. In particular we (i) establish a strong invariance principle, i.e. a quantitative coupling between a high intensity shot noise field and the Gaussian limit such that they are uniformly close on large domains with high probability, and (ii) use this to derive an asymptotic expansion for the critical level above which the excursion sets of the shot noise field percolate.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"21 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79621088","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 provide the exact large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity, improving thus a previous result on the logarithmic asymptotic behavior of this tail. Two factors influence this behavior: the distribution of the largest fragment at the time of a dislocation and the index of self-similarity. As an application we obtain the asymptotic behavior of all moments of the largest fragment and compare it to the behavior of the moments of a tagged fragment, whose decrease is in general significantly slower. We illustrate our results on several examples, including fragmentations related to random real trees - for which we thus obtain the large-time behavior of the tail distribution of the height - such as the stable L'evy trees of Duquesne, Le Gall and Le Jan (including the Brownian tree of Aldous), the alpha-model of Ford and the beta-splitting model of Aldous.
我们提供了具有负自相似指数的自相似破碎过程消光时间尾部分布的精确大时间行为，从而改进了先前关于该尾部的对数渐近行为的结果。影响这种行为的因素有两个:位错发生时最大碎片的分布和自相似指数。作为一种应用，我们得到了最大片段的所有矩的渐近行为，并将其与标记片段的矩的行为进行比较，标记片段的矩的减少通常要慢得多。我们用几个例子来说明我们的结果，包括与随机真实树相关的碎片-因此我们获得了高度尾部分布的大时间行为-例如Duquesne, Le Gall和Le Jan的稳定L'evy树(包括Aldous的布朗树)，Ford的α模型和Aldous的β分裂模型。
{"title":"Tail asymptotics for extinction times of self-similar fragmentations","authors":"Bénédicte Haas","doi":"10.1214/22-aihp1306","DOIUrl":"https://doi.org/10.1214/22-aihp1306","url":null,"abstract":"We provide the exact large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity, improving thus a previous result on the logarithmic asymptotic behavior of this tail. Two factors influence this behavior: the distribution of the largest fragment at the time of a dislocation and the index of self-similarity. As an application we obtain the asymptotic behavior of all moments of the largest fragment and compare it to the behavior of the moments of a tagged fragment, whose decrease is in general significantly slower. We illustrate our results on several examples, including fragmentations related to random real trees - for which we thus obtain the large-time behavior of the tail distribution of the height - such as the stable L'evy trees of Duquesne, Le Gall and Le Jan (including the Brownian tree of Aldous), the alpha-model of Ford and the beta-splitting model of Aldous.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"50 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81453261","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}
{"title":"Wilson loops in SYM $mathcal{N}=4$ do not parametrize an orientable space","authors":"S. Agarwala, Cameron Marcott","doi":"10.4171/aihpd/111","DOIUrl":"https://doi.org/10.4171/aihpd/111","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"94 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77033608","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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