Let X = {X(t), t ∈ ? N } be a Gaussian random field with values in ? d defined by (1) $$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$ . The properties of space and time anisotropy of X and their conne...Let X = {X(t), t ∈ ? N } be a Gaussian random field with values in ? d defined by (1) $$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$ . The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X.When X is an (N, d)-Gaussian random field as in (1), where X 1,...,X d are independent copies of a real valued, centered Gaussian random field X 0 which is anisotropic in the time variable. We establish uniform Hausdorff dimension results for the image sets of X. These results extend the corresponding results on one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet.展开更多
In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Caussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-pr...In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Caussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.展开更多
Let X={X(t)∈R^(d),t∈R^(N)}be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions.We study the packing dimension of range X(E)under the anisotropi...Let X={X(t)∈R^(d),t∈R^(N)}be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions.We study the packing dimension of range X(E)under the anisotropic(time variable)metric space(R^(N),ρ)and(space variable)metric space(R^(d),τ),where E⊂R^(N) is a Borel set.Our results generalize the corresponding results of Estrade,Wu and Xiao(Commun.Stoch.Anal.,5,41-64(2011))for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.展开更多
Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian ...Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.展开更多
This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm...This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.展开更多
We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields.It corresponds to a special case of the general situation considered in ...We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields.It corresponds to a special case of the general situation considered in Hairer and Xu(large-scale limit of interface fluctuation models.ArXiv e-prints arXiv:1802.08192,2018),but with improved estimates.As a consequence,we establish convergence of a class of Gaussian fields composite with more general functions.These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.展开更多
General limit theorems are established for l^p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l^p-value...General limit theorems are established for l^p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l^p-valued Gaussian random fields under(?)explicit conditions.展开更多
In this paper we devote ourselves to extending Berman’s sojourn time method,which is thoroughly described in[1-3],to investigate the tail asymptotics of the extrema of a Gaussian random field over[0,T]^(d) with T∈(0...In this paper we devote ourselves to extending Berman’s sojourn time method,which is thoroughly described in[1-3],to investigate the tail asymptotics of the extrema of a Gaussian random field over[0,T]^(d) with T∈(0,∞).展开更多
This paper discusses the problem of classifying a multivariate Gaussian random field observation into one of the several categories specified by different parametric mean models. Investigation is conducted on the clas...This paper discusses the problem of classifying a multivariate Gaussian random field observation into one of the several categories specified by different parametric mean models. Investigation is conducted on the classifier based on plug-in Bayes classification rule (PBCR) formed by replacing unknown parameters in Bayes classification rule (BCR) with category parameters estimators. This is the extension of the previous one from the two category cases to the multi-category case. The novel closed-form expressions for the Bayes classification probability and actual correct classification rate associated with PBCR are derived. These correct classification rates are suggested as performance measures for the classifications procedure. An empirical study has been carried out to analyze the dependence of derived classification rates on category parameters.展开更多
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covarianc...This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.展开更多
Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP>...Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m【j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.展开更多
In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its loca...In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.展开更多
Block-to-block and block-to-point kriging predictions based on block data are proposed. Blocks may be regular （mesh data） or of more general shapes. Under the assumptions of second-order stationarity and isotropicit...Block-to-block and block-to-point kriging predictions based on block data are proposed. Blocks may be regular （mesh data） or of more general shapes. Under the assumptions of second-order stationarity and isotropicity, we show how to lessen the number of calculations of relevant block-to-block or block-to-point covariances. As illustrations, a mesh data of population and a simulated block data on convex polygons are analyzed.展开更多
This paper proposes a semi-supervised inductive algorithm adopting a Gaussian random field(GRF)and Gaussian process.We introduce the prior based on graph regularization.This regularization term measures the p-smoothne...This paper proposes a semi-supervised inductive algorithm adopting a Gaussian random field(GRF)and Gaussian process.We introduce the prior based on graph regularization.This regularization term measures the p-smoothness over the graph.A new conditional probability called the extended Bernoulli model(EBM)is also proposed.EBM generalizes the logistic regression to the semi-supervised case,and especially,it can naturally represent the margin.In the training phase,a novel solution is given to the discrete regularization framework defined on the graphs.For the new test data,we present the prediction formulation,and explain how the margin model affects the classification boundary.A hyper-parameter estimation method is also developed.Experimental results show that our method is competitive with the existing semi-supervised inductive and transductive methods.展开更多
In this paper,we propose a novel approach to the multiple-sample testing problem using a studentized test statistic based on the random lifter technique.The method reformulates the classical Ksample test as an indepen...In this paper,we propose a novel approach to the multiple-sample testing problem using a studentized test statistic based on the random lifter technique.The method reformulates the classical Ksample test as an independence test between two random variables,enabling more efficient handling of complex data types.We solve problems with the non-standard normal limiting distributions of degenerate U-statistics using a random lifter approach.This creates a test statistic that is asymptotically normal under the null hypothesis.Numerous simulations and real-world applications have demonstrated that our method performs well with many data types,including Euclidean,directional,and symmetric positive definite data.It is also very good at controlling Type I errors.Our method also shows significant computational efficiency,outperforming existing K-sample tests,particularly when applied to large datasets.These results suggest that the proposed method is a powerful and practical solution for multiple-sample testing in complex data scenarios.展开更多
In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t...In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.展开更多
基金supported by National Science Foundation of the United States (Grant No.DMS-0706728)
文摘Let X = {X(t), t ∈ ? N } be a Gaussian random field with values in ? d defined by (1) $$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$ . The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X.When X is an (N, d)-Gaussian random field as in (1), where X 1,...,X d are independent copies of a real valued, centered Gaussian random field X 0 which is anisotropic in the time variable. We establish uniform Hausdorff dimension results for the image sets of X. These results extend the corresponding results on one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet.
基金Project supported by the National Natural Science Foundation of China(No.11071182)
文摘In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Caussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.
基金Supported by National Natural Science Foundation of China(Grant No.11971432)Natural Science Foundation of Zhejiang Province(Grant No.LY21G010003)+2 种基金Humanities and Social Sciences Foundation of the Ministry of Education(Grant No.18YJA910001)First Class Discipline of Zhejiang-A(Zhejiang Gongshang University-Statistics)Natural Science Foundation of Chuzhou University(Grant No.zrjz2019012)。
文摘Let X={X(t)∈R^(d),t∈R^(N)}be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions.We study the packing dimension of range X(E)under the anisotropic(time variable)metric space(R^(N),ρ)and(space variable)metric space(R^(d),τ),where E⊂R^(N) is a Borel set.Our results generalize the corresponding results of Estrade,Wu and Xiao(Commun.Stoch.Anal.,5,41-64(2011))for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.
基金Supported by the National Natural Science Foundation of China （No.10471148）, Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei （No.2003A005）Acknowledgements. We wish to express our sincere thanks to Professor Xiao Yimin for suggesting the problem to me and for his subsequent encouragement and help.
文摘Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.
基金Supported by NSFC(Grants Nos.11671115,11731012 and 11871425)NSF(Grant No.DMS-1855185)
文摘This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.
基金the support from the Engineering and Physical Sciences Research Council through the fellowship EP/N021568/1.
文摘We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields.It corresponds to a special case of the general situation considered in Hairer and Xu(large-scale limit of interface fluctuation models.ArXiv e-prints arXiv:1802.08192,2018),but with improved estimates.As a consequence,we establish convergence of a class of Gaussian fields composite with more general functions.These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.
基金This work was supported by NSERC Canada grants at Carleton University and by KOSEF-R01-2005-000-10696-0
文摘General limit theorems are established for l^p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l^p-valued Gaussian random fields under(?)explicit conditions.
基金partially supported by National Natural Science Foundation of China(11701070,71871046)Ronglian Scholarship Fund.
文摘In this paper we devote ourselves to extending Berman’s sojourn time method,which is thoroughly described in[1-3],to investigate the tail asymptotics of the extrema of a Gaussian random field over[0,T]^(d) with T∈(0,∞).
文摘This paper discusses the problem of classifying a multivariate Gaussian random field observation into one of the several categories specified by different parametric mean models. Investigation is conducted on the classifier based on plug-in Bayes classification rule (PBCR) formed by replacing unknown parameters in Bayes classification rule (BCR) with category parameters estimators. This is the extension of the previous one from the two category cases to the multi-category case. The novel closed-form expressions for the Bayes classification probability and actual correct classification rate associated with PBCR are derived. These correct classification rates are suggested as performance measures for the classifications procedure. An empirical study has been carried out to analyze the dependence of derived classification rates on category parameters.
基金supported by an NSERC granta startup fund of University of Alberta
文摘This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
基金NSERC Canada grants of Miklos Csorgo and Barbara Szyszkowicz at Carleton University,Ottawa,and by KRF-2003-C00098NSERC Canada grants at Carleton University,Ottawa
文摘Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m【j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.
基金supported by the National Natural Science Foundation of China (No. 10871177)the Ph. D.Programs Foundation of Ministry of Education of China (No. 20060335032)the Natural Science Foundation of Zhejiang Province of China (No. Y7080044)
文摘In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.
文摘Block-to-block and block-to-point kriging predictions based on block data are proposed. Blocks may be regular （mesh data） or of more general shapes. Under the assumptions of second-order stationarity and isotropicity, we show how to lessen the number of calculations of relevant block-to-block or block-to-point covariances. As illustrations, a mesh data of population and a simulated block data on convex polygons are analyzed.
基金This work was supported by the Basic Research Foundation of Tsinghua National Laboratory for Information Science and Technology(TNList).
文摘This paper proposes a semi-supervised inductive algorithm adopting a Gaussian random field(GRF)and Gaussian process.We introduce the prior based on graph regularization.This regularization term measures the p-smoothness over the graph.A new conditional probability called the extended Bernoulli model(EBM)is also proposed.EBM generalizes the logistic regression to the semi-supervised case,and especially,it can naturally represent the margin.In the training phase,a novel solution is given to the discrete regularization framework defined on the graphs.For the new test data,we present the prediction formulation,and explain how the margin model affects the classification boundary.A hyper-parameter estimation method is also developed.Experimental results show that our method is competitive with the existing semi-supervised inductive and transductive methods.
基金supported by National Natural Science Foundation of China(Grant Nos.12231017,72171216,71921001 and 71991474)the National Key R&D Program of China(Grant No.2022YFA1003803)supported by National Natural Science Foundation of China(Grant Nos.71873128 and 72293573).
文摘In this paper,we propose a novel approach to the multiple-sample testing problem using a studentized test statistic based on the random lifter technique.The method reformulates the classical Ksample test as an independence test between two random variables,enabling more efficient handling of complex data types.We solve problems with the non-standard normal limiting distributions of degenerate U-statistics using a random lifter approach.This creates a test statistic that is asymptotically normal under the null hypothesis.Numerous simulations and real-world applications have demonstrated that our method performs well with many data types,including Euclidean,directional,and symmetric positive definite data.It is also very good at controlling Type I errors.Our method also shows significant computational efficiency,outperforming existing K-sample tests,particularly when applied to large datasets.These results suggest that the proposed method is a powerful and practical solution for multiple-sample testing in complex data scenarios.
基金Supported by the National Natural Science Foundation of China(No.10571159)Specialized Research Fund for the Doctor Program of Higher Education(No.2002335090)
文摘In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.
