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Random Lipschitz–Killing curvatures: Reduction Principles, Integration by Parts and Wiener chaos 随机Lipschitz-Killing曲率:约简原理,分部积分和Wiener混沌
IF 0.9 Q3 Mathematics Pub Date : 2021-08-03 DOI: 10.1090/tpms/1170
Anna Vidotto
In this survey we collect some recent results regarding the Lipschitz–Killing curvatures (LKCs) of the excursion sets of random eigenfunctions on the two-dimensional standard flat torus (arithmetic random waves) and on the two-dimensional unit sphere (random spherical harmonics). In particular, the aim of the present survey is to highlight the key role of integration by parts formulae in order to have an extremely neat expression for the random LKCs. Indeed, the main tool to study local geometric functionals of random waves on manifold is to exploit their Wiener chaos decomposition and show that (often), in the so-called high-energy limit, a single chaotic component dominates their behavior. Moreover, reduction principles show that the dominant Wiener chaotic component of LKCs of random waves’ excursion sets at threshold level u ≠ 0 une 0 is proportional to the integral of H 2 ( f ) H_2(f) , f f being the random field of interest and H 2 H_2 the second Hermite polynomial. This will be shown via integration by parts formulae.
本文收集了二维标准平面(算术随机波)和二维单位球(随机球谐波)上随机特征函数漂移集的Lipschitz-Killing曲率的一些最新结果。特别地,本调查的目的是强调分部积分公式的关键作用,以便对随机LKCs有一个非常整洁的表达式。实际上,研究流形上随机波的局部几何泛函的主要工具是利用它们的维纳混沌分解,并表明(通常)在所谓的高能极限下,一个单一的混沌分量支配着它们的行为。此外,约简原理表明,在阈值水平u≠0 u ne0处随机波漂移集LKCs的优势Wiener混沌分量与h2 (f) H_2(f)的积分成正比,其中f为感兴趣的随机场,h2 H_2为第二个Hermite多项式。这将通过分部积分公式来展示。
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引用次数: 2
On the correlation between critical points and critical values for random spherical harmonics 随机球谐波的临界点与临界值的相关性
IF 0.9 Q3 Mathematics Pub Date : 2021-07-31 DOI: 10.1090/tpms/1164
Valentina Cammarota, Anna Todino
We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval I ⊂ R I subset mathbb {R} . We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random L 2 L^2 -norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics.
我们研究了随机球谐的临界点总数与任意区间I⊂R Isubet mathbb{R}中有值的临界点数量之间的相关性。我们证明了相关是渐近零的,而偏相关在控制了本征函数球面上的随机L2L^2-范数后,是渐近一的。我们的发现补充了Wigman(2012)、Marinucci和Rossi(2021)关于随机球面谐波的节点长度和边界长度之间相关性的结果。
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引用次数: 0
Parametric estimation for functional autoregressive processes on the sphere 球面上函数自回归过程的参数估计
IF 0.9 Q3 Mathematics Pub Date : 2021-07-19 DOI: 10.1090/tpms/1165
Alessia Caponera, C. Durastanti
The aim of this paper is to define a nonlinear least squares estimator for the spectral parameters of a spherical autoregressive process of order 1 in a parametric setting. Furthermore, we investigate on its asymptotic properties, such as weak consistency and asymptotic normality.
本文的目的是为参数设置中的1阶球面自回归过程的谱参数定义一个非线性最小二乘估计量。此外,我们还研究了它的渐近性质,如弱一致性和渐近正态性。
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引用次数: 0
Fractional stochastic partial differential equation for random tangent fields on the sphere 球面上随机切线场的分数阶随机偏微分方程
IF 0.9 Q3 Mathematics Pub Date : 2021-07-08 DOI: 10.1090/tpms/1142
V. Anh, A. Olenko, Yu Guang Wang
This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the Lévy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Loève expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. The variance of the increments and approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.
本文发展了一个分数阶随机偏微分方程(SPDE)来模拟单位球面上随机切向量场的演化。SPDE由分数扩散算子控制,以对空间解的Lévy型行为建模,时间上的分数导数描述其时间解的间歇性,并由单位球面上的向量值分数布朗运动驱动,以表征其时间-长程依赖性。SPDE的解以向量球谐波的Karhunen-Loève展开形式给出。它的协方差矩阵函数被建立为单位球面上的张量场,该张量场是勒让德张量核的扩展。研究了解的增量和近似的方差,给出了近似误差的收敛速度。证明了这些收敛速度如何取决于分数布朗运动的功率谱衰减和方差。
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引用次数: 4
A test on mean-variance efficiency of the tangency portfolio in high-dimensional setting 高维环境下切线投资组合的均方差效率检验
IF 0.9 Q3 Mathematics Pub Date : 2021-06-16 DOI: 10.1090/TPMS/1136
Stanislas Muhinyuza
In this paper we derive the asymptotic distribution of the test of the efficiency of the tangency portfolio in high-dimensional settings, namely when both the portfolio dimension and the sample siz ...
本文导出了高维环境下切线投资组合效率检验的渐近分布,即当投资组合维数和样本量同时存在时。
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引用次数: 2
A note on last-success-problem 关于last-success-problem的注释
IF 0.9 Q3 Mathematics Pub Date : 2021-06-16 DOI: 10.1090/TPMS/1139
J. M. G. Ribas
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引用次数: 2
Editorial 编辑
IF 0.9 Q3 Mathematics Pub Date : 2021-06-16 DOI: 10.1090/tpms/1140
Y. Mishura, L. Sakhno, A. Veretennikov
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引用次数: 0
Closed-form estimator for the matrix-variate Gamma distribution 矩阵变量伽玛分布的封闭估计量
IF 0.9 Q3 Mathematics Pub Date : 2021-06-16 DOI: 10.1090/TPMS/1138
Gustav Alfelt
In this paper we present a novel closed-form estimator for the parameters of the matrixvariate gamma distribution. The estimator relies on the moments of a transformation of the observed matrices, and is compared to the maximum likelihood estimator (MLE) through a simulation study. The study reveals that the suggested estimator outperforms the MLE, in terms of estimation error, when the underlying scale matrix parameter is ill-conditioned or when the shape parameter is close to its lower bound. In addition, since the suggested estimator is closed-form, it does not require numerical optimization as the MLE does, thus needing shorter computation time and is furthermore not subject to start value sensitivity or convergence issues. Finally, using the proposed estimator as start value in the optimization procedure of the MLE is shown to substantially reduce computation time, in comparison to using arbitrary start values.
本文给出了矩阵变量分布参数的一种新的闭型估计量。该估计量依赖于观测矩阵变换的矩量,并通过仿真研究与极大似然估计量(MLE)进行了比较。研究表明,当底层尺度矩阵参数为病态或形状参数接近其下界时,所提出的估计器在估计误差方面优于MLE。此外,由于建议的估计器是闭型的,因此不需要像MLE那样进行数值优化,因此需要更短的计算时间,并且不受起始值敏感性或收敛性问题的影响。最后,与使用任意起始值相比,在MLE优化过程中使用所提出的估计量作为起始值大大减少了计算时间。
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引用次数: 1
On modeling the correlation as an additional parameter in random effects model 随机效应模型中作为附加参数的相关性建模
IF 0.9 Q3 Mathematics Pub Date : 2021-06-16 DOI: 10.1090/TPMS/1137
Rebecca Nalule Muhumuza, Olha Bodnar
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引用次数: 0
Asymptotic results for certain first-passage times and areas of renewal processes 更新过程的某些首次通过时间和区域的渐近结果
IF 0.9 Q3 Mathematics Pub Date : 2021-05-17 DOI: 10.1090/tpms/1189
C. Macci, B. Pacchiarotti

We consider the process { x N ( t ) : t 0 } {x-N(t):tgeq 0} , where x R + xin mathbb {R}_+ and { N ( t ) : t 0 } {N(t):tgeq 0} is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of ( τ ( x ) , A ( x )

我们考虑过程{x−N(t):t≥0}{x-N(t):tgeq0},其中x∈R+xinmathbb{R}_+并且{N(t):t≥0}{N(t):tgeq0}是具有轻尾分布保持时间的更新过程。我们感兴趣的是(τ(x),A(x)),其中τ,和A(x)≔õ0τ(x)(x−N(t))d t A(x:t≥0}{x-N(t):tgeq 0}。我们注意到,通过引用积分随机游动的概念,我们可以定义序列{(τ(n),A(n)):n≥1}{(tau(n)、A(n)):ngeq1}。我们的目的是证明x的渐近结果→ ∞ x以大(和中等)偏差的方式存在。
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引用次数: 0
期刊
Theory of Probability and Mathematical Statistics
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