Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) {M(t)}t∈R{{mathcal {M}}(t)}_{tin mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H{mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H(t){mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multi
{"title":"On local path behavior of Surgailis multifractional processes","authors":"A. Ayache, F. Bouly","doi":"10.1090/tpms/1162","DOIUrl":"https://doi.org/10.1090/tpms/1162","url":null,"abstract":"<p>Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the <italic>classical</italic> Multifractional Brownian Motion (MBM) <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:msub>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{{mathcal {M}}(t)}_{tin mathbb {R}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{mathcal {H}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of the well-known Fractional Brownian Motion by a deterministic function <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H left-parenthesis t right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{mathcal {H}}(t)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two <italic>non-classical</italic> Gaussian multi","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45008599","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}
Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields Z(x)Z(x) and construct approximations of solutions U(x)U(x) of ordinary or partial differential equations whose random coefficients depend on Z(x)Z(x). FD models of Z(x)Z(x) and U(x)U(x) constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of
{"title":"Finite dimensional models for random microstructures","authors":"M. Grigoriu","doi":"10.1090/tpms/1168","DOIUrl":"https://doi.org/10.1090/tpms/1168","url":null,"abstract":"<p>Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Z</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and construct approximations of solutions <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">U(x)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of ordinary or partial differential equations whose random coefficients depend on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Z</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. FD models of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Z</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">U(x)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthes","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47711515","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 analyze the limit behavior in distribution of the spatial average of the solution to the wave equation driven by the two-parameter Rosenblatt process in spatial dimension d = 1 d=1 . We prove that this spatial average satisfies a non-central limit theorem, more precisely it converges in law to a Wiener integral with respect to the Rosenblatt process. We also give a functional version of this limit theorem.
{"title":"Non-central limit theorem for the spatial average of the solution to the wave equation with Rosenblatt noise","authors":"R. Dhoyer, C. Tudor","doi":"10.1090/tpms/1167","DOIUrl":"https://doi.org/10.1090/tpms/1167","url":null,"abstract":"We analyze the limit behavior in distribution of the spatial average of the solution to the wave equation driven by the two-parameter Rosenblatt process in spatial dimension \u0000\u0000 \u0000 \u0000 d\u0000 =\u0000 1\u0000 \u0000 d=1\u0000 \u0000\u0000. We prove that this spatial average satisfies a non-central limit theorem, more precisely it converges in law to a Wiener integral with respect to the Rosenblatt process. We also give a functional version of this limit theorem.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46663930","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 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":" ","pages":""},"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":"1 1","pages":""},"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":" ","pages":""},"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}
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":" ","pages":""},"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}
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":" ","pages":""},"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}
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