The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings. We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.
{"title":"Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus","authors":"F. Bachoc, A. Peron, E. Porcu","doi":"10.1090/tpms/1176","DOIUrl":"https://doi.org/10.1090/tpms/1176","url":null,"abstract":"The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings.\u0000\u0000We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47166409","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}
Boltzmann–Gibbs random fields are defined in terms of the exponential expression exp ( − H ) exp left (-mathcal {H}right ) , where H mathcal {H} is a suitably defined energy functional of the field states x ( s ) x(mathbf {s}) . This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with ν = 1 nu =1 are established.
{"title":"Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics","authors":"D. Hristopulos","doi":"10.1090/tpms/1180","DOIUrl":"https://doi.org/10.1090/tpms/1180","url":null,"abstract":"Boltzmann–Gibbs random fields are defined in terms of the exponential expression \u0000\u0000 \u0000 \u0000 exp\u0000 \u0000 \u0000 (\u0000 −\u0000 \u0000 H\u0000 \u0000 )\u0000 \u0000 \u0000 exp left (-mathcal {H}right )\u0000 \u0000\u0000, where \u0000\u0000 \u0000 \u0000 H\u0000 \u0000 mathcal {H}\u0000 \u0000\u0000 is a suitably defined energy functional of the field states \u0000\u0000 \u0000 \u0000 x\u0000 (\u0000 \u0000 s\u0000 \u0000 )\u0000 \u0000 x(mathbf {s})\u0000 \u0000\u0000. This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with \u0000\u0000 \u0000 \u0000 ν\u0000 =\u0000 1\u0000 \u0000 nu =1\u0000 \u0000\u0000 are established.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42039987","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 introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.
{"title":"Series representations and simulations of isotropic random fields in the Euclidean space","authors":"Z. Ma, C. Ma","doi":"10.1090/tpms/1158","DOIUrl":"https://doi.org/10.1090/tpms/1158","url":null,"abstract":"This paper introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41682804","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 for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.
{"title":"On the locations of maxima and minima in a sequence of exchangeable random variables","authors":"D. Ferger","doi":"10.1090/tpms/1154","DOIUrl":"https://doi.org/10.1090/tpms/1154","url":null,"abstract":"We show for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49260411","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 Xmathbb {X} and Ymathbb {Y} be two complete, separable, metric spaces, ξε(x),x∈Xxi _varepsilon (x), x in mathbb {X} and νεnu _varepsilon be, for every ε∈[0,1]varepsilon in [0, 1], respectively, a random field taking values in space Y
A multivariate trigonometric regression model is considered. Various discrete modifications of the similar bivariate model received serious attention in the literature on signal and image processing due to multiple applications in the analysis of symmetric textured surfaces. In the paper asymptotic normality of the least squares estimator for amplitudes and angular frequencies is obtained in multivariate trigonometric model assuming that the random noise is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly dependent random field on R M , M > 2. mathbb {R}^M,,, M>2.
{"title":"On the least squares estimator asymptotic normality of the multivariate symmetric textured surface parameters","authors":"A. Ivanov, I. Savych","doi":"10.1090/tpms/1161","DOIUrl":"https://doi.org/10.1090/tpms/1161","url":null,"abstract":"A multivariate trigonometric regression model is considered. Various discrete modifications of the similar bivariate model received serious attention in the literature on signal and image processing due to multiple applications in the analysis of symmetric textured surfaces. In the paper asymptotic normality of the least squares estimator for amplitudes and angular frequencies is obtained in multivariate trigonometric model assuming that the random noise is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly dependent random field on \u0000\u0000 \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 M\u0000 \u0000 ,\u0000 \u0000 \u0000 M\u0000 >\u0000 2.\u0000 \u0000 mathbb {R}^M,,, M>2.\u0000 \u0000\u0000","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48147833","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 X=(Xt)t≥0X=(X_t)_{tgeq 0} be a one-dimensional Lévy process such that each XtX_t has a Cb1C^1_b-density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f:R→Rfcolon mathbb {R}to mathbb {R}, and exponentially bounded functions g:R→
We prove a functional limit theorem for the number of visits by a planar random walk on Z 2 mathbb {Z}^2 with zero mean and finite second moment to the points of a fixed finite set P ⊂ Z 2 Psubset mathbb {Z}^2 . The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.
{"title":"On the local time of a recurrent random walk on ℤ²","authors":"V. Bohun, A. Marynych","doi":"10.1090/tpms/1156","DOIUrl":"https://doi.org/10.1090/tpms/1156","url":null,"abstract":"We prove a functional limit theorem for the number of visits by a planar random walk on \u0000\u0000 \u0000 \u0000 \u0000 Z\u0000 \u0000 2\u0000 \u0000 mathbb {Z}^2\u0000 \u0000\u0000 with zero mean and finite second moment to the points of a fixed finite set \u0000\u0000 \u0000 \u0000 P\u0000 ⊂\u0000 \u0000 \u0000 Z\u0000 \u0000 2\u0000 \u0000 \u0000 Psubset mathbb {Z}^2\u0000 \u0000\u0000. The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44724360","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 discuss joint spatial-temporal scaling limits of sums Aλ,γA_{lambda ,gamma } (indexed by (x,y)∈R+2(x,y) in mathbb {R}^2_+) of large number O(λγ)O(lambda ^{gamma }) of independent copies of integrated input process X={X(t),t∈R}X = {X(t), t in mathbb {R}}
This paper studies the behaviour of quadratic variations of a stochastic wave equation driven by a noise that is white in space and fractional in time. Complementing the analysis of quadratic variations in the space component carried out in [Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise, Electron. J. Stat. 12 (2018), no. 2, 3639–3672] and [Generalized k k -variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180], it focuses on the time component of the solution process. For different values of the Hurst parameter a central and a noncentral limit theorems are proved, allowing to construct parameter estimators and compare them to the findings in the space-dependent case. Finally, rectangular quadratic variations in the white noise case are studied and a central limit theorem is demonstrated.
{"title":"On quadratic variations for the fractional-white wave equation","authors":"Radomyra Shevchenko","doi":"10.1090/tpms/1192","DOIUrl":"https://doi.org/10.1090/tpms/1192","url":null,"abstract":"This paper studies the behaviour of quadratic variations of a stochastic wave equation driven by a noise that is white in space and fractional in time. Complementing the analysis of quadratic variations in the space component carried out in [Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise, Electron. J. Stat. 12 (2018), no. 2, 3639–3672] and [Generalized \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180], it focuses on the time component of the solution process. For different values of the Hurst parameter a central and a noncentral limit theorems are proved, allowing to construct parameter estimators and compare them to the findings in the space-dependent case. Finally, rectangular quadratic variations in the white noise case are studied and a central limit theorem is demonstrated.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47831804","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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