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。
{"title":"Random Lipschitz–Killing curvatures: Reduction Principles, Integration by Parts and Wiener chaos","authors":"Anna Vidotto","doi":"10.1090/tpms/1170","DOIUrl":"https://doi.org/10.1090/tpms/1170","url":null,"abstract":"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 \u0000\u0000 \u0000 \u0000 u\u0000 ≠\u0000 0\u0000 \u0000 une 0\u0000 \u0000\u0000 is proportional to the integral of \u0000\u0000 \u0000 \u0000 \u0000 H\u0000 2\u0000 \u0000 (\u0000 f\u0000 )\u0000 \u0000 H_2(f)\u0000 \u0000\u0000, \u0000\u0000 \u0000 f\u0000 f\u0000 \u0000\u0000 being the random field of interest and \u0000\u0000 \u0000 \u0000 H\u0000 2\u0000 \u0000 H_2\u0000 \u0000\u0000 the second Hermite polynomial. This will be shown via integration by parts formulae.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41660632","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 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.
{"title":"On the correlation between critical points and critical values for random spherical harmonics","authors":"Valentina Cammarota, Anna Todino","doi":"10.1090/tpms/1164","DOIUrl":"https://doi.org/10.1090/tpms/1164","url":null,"abstract":"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 \u0000\u0000 \u0000 \u0000 I\u0000 ⊂\u0000 \u0000 R\u0000 \u0000 \u0000 I subset mathbb {R}\u0000 \u0000\u0000. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random \u0000\u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 L^2\u0000 \u0000\u0000-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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46762967","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 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.
{"title":"Parametric estimation for functional autoregressive processes on the sphere","authors":"Alessia Caponera, C. Durastanti","doi":"10.1090/tpms/1165","DOIUrl":"https://doi.org/10.1090/tpms/1165","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49112399","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 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.
{"title":"Fractional stochastic partial differential equation for random tangent fields on the sphere","authors":"V. Anh, A. Olenko, Yu Guang Wang","doi":"10.1090/tpms/1142","DOIUrl":"https://doi.org/10.1090/tpms/1142","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47164807","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 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。
{"title":"A test on mean-variance efficiency of the tangency portfolio in high-dimensional setting","authors":"Stanislas Muhinyuza","doi":"10.1090/TPMS/1136","DOIUrl":"https://doi.org/10.1090/TPMS/1136","url":null,"abstract":"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 ...","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45020288","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":"A note on last-success-problem","authors":"J. M. G. Ribas","doi":"10.1090/TPMS/1139","DOIUrl":"https://doi.org/10.1090/TPMS/1139","url":null,"abstract":"","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45259958","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":"Editorial","authors":"Y. Mishura, L. Sakhno, A. Veretennikov","doi":"10.1090/tpms/1140","DOIUrl":"https://doi.org/10.1090/tpms/1140","url":null,"abstract":"","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45884285","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 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.
{"title":"Closed-form estimator for the matrix-variate Gamma distribution","authors":"Gustav Alfelt","doi":"10.1090/TPMS/1138","DOIUrl":"https://doi.org/10.1090/TPMS/1138","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45223114","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":"On modeling the correlation as an additional parameter in random effects model","authors":"Rebecca Nalule Muhumuza, Olha Bodnar","doi":"10.1090/TPMS/1137","DOIUrl":"https://doi.org/10.1090/TPMS/1137","url":null,"abstract":"","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43289468","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 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。
{"title":"Asymptotic results for certain first-passage times and areas of renewal processes","authors":"C. Macci, B. Pacchiarotti","doi":"10.1090/tpms/1189","DOIUrl":"https://doi.org/10.1090/tpms/1189","url":null,"abstract":"<p>We consider the process <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet x minus upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{x-N(t):tgeq 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x element-of double-struck upper R Subscript plus\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">xin mathbb {R}_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{N(t):tgeq 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis tau left-parenthesis x right-parenthesis comma upper A left-parenthesis x right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"f","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49188078","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：