It is proposed a class of statistical estimators H = (H1,… ,Hd) for the Hurst parameters H = (H1,… ,Hd) of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotic...It is proposed a class of statistical estimators H = (H1,… ,Hd) for the Hurst parameters H = (H1,… ,Hd) of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging (DTI) of nuclear magnetic resonance (NMR). Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when Hi ≥ 1/2, the estimators are accurate, and when Hi 〈 1/2, there are some bias.展开更多
Let B^H={B^H(t),t∈R^N+}be a real-valued(N,d)fractional Brownian sheet with Hurst index H=(H1,…,HN).The characteristics of the polar functions for B^H are discussed.The relationship between the class of contin...Let B^H={B^H(t),t∈R^N+}be a real-valued(N,d)fractional Brownian sheet with Hurst index H=(H1,…,HN).The characteristics of the polar functions for B^H are discussed.The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of B^H is obtained.The Hausdorff dimension about the fixed points and the inequality about the Kolmogorov’s entropy index for B^H are presented.Furthermore,it is proved that any two independent fractional Brownian sheets are nonintersecting in some conditions.A problem proposed by LeGall about the existence of no-polar continuous functions satisfying the Holder condition is also solved.展开更多
Let B^H'K={B^H'K(t), t∈R+^N} be an (N,d)-bifractional Brownian sheet with Hurst indices H = (H1,…,HN) C∈0,1)^N and K = (K1,…,KN) ∈ (0,1]^N. The properties of the polar sets of B^H'K are discussed. T...Let B^H'K={B^H'K(t), t∈R+^N} be an (N,d)-bifractional Brownian sheet with Hurst indices H = (H1,…,HN) C∈0,1)^N and K = (K1,…,KN) ∈ (0,1]^N. The properties of the polar sets of B^H'K are discussed. The sufficient conditions and necessary conditions for a compact set to be polar for B^H'K are proved. The infimum of Hausdorff dimensions of its non-polar sets are obtained by means of constructing a Cantor-type set to connect its Hausdorff dimension and capacity.展开更多
Let W={W(t);t ∈R_+~N} be the d-dimensional N-parameter Brownian Sheet. Sufficient conditions for a compact set F C Rd \ {0} to be a polar set for W are proved. It is also proved that if 2N≤d, then for any compact se...Let W={W(t);t ∈R_+~N} be the d-dimensional N-parameter Brownian Sheet. Sufficient conditions for a compact set F C Rd \ {0} to be a polar set for W are proved. It is also proved that if 2N≤d, then for any compact set E (?)R_>~N ,inf {dimF : F ∈B(Rd), P{W(E) ∩F≠(?)}>0} = d- 2DiroE, and if 2N > d, then for any compact set F C Rd \ {0}, inf{dim E : E ∈ B(R_>~N), P{W(E)∩F≠(?)}>0}=d/2-DimF/2,where B(Rd) and B(R_>~N) denote the Borel σ-algebra in Rd and R_>~N respectively, and dim and Dim are Hausdorff dimension and Packing dimension respectively.展开更多
Let W^~ be a two-parameter, R^d-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under W^~. He also presents the connections betw...Let W^~ be a two-parameter, R^d-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under W^~. He also presents the connections between the Lebesgue measure of the image of W^~ and Bessel-Riesz capacity. His conclusions also solve a problem proposed by J.- P.Kahane.展开更多
Let W^-(t)(t∈R+^N) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-(t). It is proved that for any α∈ R^d, P{W^-(t) = α, for some t∈ R〉^N} = {1, if βd 〈 2N ...Let W^-(t)(t∈R+^N) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-(t). It is proved that for any α∈ R^d, P{W^-(t) = α, for some t∈ R〉^N} = {1, if βd 〈 2N ,0 if αd〉 2N and the probability that W^-(t) has k-multiple points is 1 or 0 according as whether 2kN〉d(k-1)β or 2kN 〈 d(k - 1)α. These results contain and extend the results of the Brownian sheet, where R〉^N = (0,+∞)U,R+^N = [0,+∞)^N,0〈 α ≤1and β〉1.展开更多
Let B^H,K : (B^H,K(t), t ∈R+^N} be an (N,d)-bifractional Brownian sheet with Hurst indices H = (H1,..., HN) ∈ (0, 1)^N and K = (K1,..., KN)∈ (0, 1]^N. The characteristics of the polar functions for B^...Let B^H,K : (B^H,K(t), t ∈R+^N} be an (N,d)-bifractional Brownian sheet with Hurst indices H = (H1,..., HN) ∈ (0, 1)^N and K = (K1,..., KN)∈ (0, 1]^N. The characteristics of the polar functions for B^H,K are investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of B^H,K is presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov's entropy index for B^H,K are obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.展开更多
Let W^~=^~{W^~(t); t∈ R+^N) be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for (t,W^~(t)) are proved. It is a...Let W^~=^~{W^~(t); t∈ R+^N) be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for (t,W^~(t)) are proved. It is also proved that if 2N ≤αd, then for any compact set E ∩→ R〉^N,d-2/α Dim E≤inf{dim F:F∈B(R^d),P{W^~(E)∩F≠0}〉0}≤d-2/βDimE,and if 2N〉αd, then for any compact set F∪→R^d/{0},α/2(d-DimF)≤inf{dimE:E∈B(R〉^N),P{W^~(E)∩F≠0}〉0}≤β/2(d-DimF),where B(R^d) and B(R〉^N) denote the Borel σ-algebra in R^d and in R〉^N respectively, dim and Dim are Hausdorff dimension and Packing dimension respectively.展开更多
For the N-parameter d-dimensional Generalized Brownian Sheet (in short G. B. S.) satisfying some conditions, this paper proves that when 2N≤αd, with probability one, E∈B(R_+~N), there holds 2/βdim E+2N-2Nβ/α≤d...For the N-parameter d-dimensional Generalized Brownian Sheet (in short G. B. S.) satisfying some conditions, this paper proves that when 2N≤αd, with probability one, E∈B(R_+~N), there holds 2/βdim E+2N-2Nβ/α≤dim (E)≤2/αdim E. Especially when α=β=1, we have dim (E)=2 dim E. Noting that even if α=β=1, the G. B. S. is wider than the Brownian Sheet, thus we have extended the uniform dimension result of Mountford, T,S.展开更多
Let B0^H = {B0^H(t),t ∈ R+^N) be a real-valued fractional Brownian sheet. Define the (N,d)- Gaussian random field B^H by B^H(t) = (B1^H(t),...,Bd^H(t)) t ∈ R+^N, where B1^H, ..., Bd^H are independent...Let B0^H = {B0^H(t),t ∈ R+^N) be a real-valued fractional Brownian sheet. Define the (N,d)- Gaussian random field B^H by B^H(t) = (B1^H(t),...,Bd^H(t)) t ∈ R+^N, where B1^H, ..., Bd^H are independent copies of B0^H. The existence and joint continuity of local times of B^H is proven in some given conditions in [22]. We then study further properties of the local times of B^H, such as the moments of increments of local times, the large increments and the maximum moduli of continuity of local times and as a result, we answer the questions posed in [22].展开更多
Monte Carlo and quasi-Monte Carlomethods arewidely used in scientific studies.As quasi-Monte Carlo simulations have advantage over ordinaryMonte Carlomethods,this paper proposes a new quasi-Monte Carlo method to simul...Monte Carlo and quasi-Monte Carlomethods arewidely used in scientific studies.As quasi-Monte Carlo simulations have advantage over ordinaryMonte Carlomethods,this paper proposes a new quasi-Monte Carlo method to simulate Brownian sheet via its Karhunen–Loéve expansion.The proposed new approach allocates quasi-random sequences for the simulation of random components of the Karhunen–Loéve expansion by maximum reducing its variability.We apply the quasi-MonteCarlo approach to an option pricing problem for a class of interest rate models whose instantaneous forward rate driven by a different stochastic shock through Brownian sheet andwe demonstrate the application with an empirical problem.展开更多
In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and s...In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and small deviation for a Brownian sheet.展开更多
基金supported in part by the National Basic Research Program of China(973 Program,2013CB910200,and 2011CB707802)
文摘It is proposed a class of statistical estimators H = (H1,… ,Hd) for the Hurst parameters H = (H1,… ,Hd) of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging (DTI) of nuclear magnetic resonance (NMR). Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when Hi ≥ 1/2, the estimators are accurate, and when Hi 〈 1/2, there are some bias.
基金the Key Research Base for Humanities and Social Sciences of Zhejiang Provincial High Education Talents(Statistics of Zhejiang Gongshang University)the Natural ScienceFoundation of Shaanxi Province(2005A08,2006A14)
文摘Let B^H={B^H(t),t∈R^N+}be a real-valued(N,d)fractional Brownian sheet with Hurst index H=(H1,…,HN).The characteristics of the polar functions for B^H are discussed.The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of B^H is obtained.The Hausdorff dimension about the fixed points and the inequality about the Kolmogorov’s entropy index for B^H are presented.Furthermore,it is proved that any two independent fractional Brownian sheets are nonintersecting in some conditions.A problem proposed by LeGall about the existence of no-polar continuous functions satisfying the Holder condition is also solved.
基金supported by the national natural foundationof China (70871104)the key research base for humanities and social sciences of Zhejiang Provincial high education talents (Statistics of Zhejiang Gongshang University)
文摘Let B^H'K={B^H'K(t), t∈R+^N} be an (N,d)-bifractional Brownian sheet with Hurst indices H = (H1,…,HN) C∈0,1)^N and K = (K1,…,KN) ∈ (0,1]^N. The properties of the polar sets of B^H'K are discussed. The sufficient conditions and necessary conditions for a compact set to be polar for B^H'K are proved. The infimum of Hausdorff dimensions of its non-polar sets are obtained by means of constructing a Cantor-type set to connect its Hausdorff dimension and capacity.
基金Supported by Sci-tech Innovation Project of Educational Department of Hubei ProvinceMajor Project of Educational Department of Hubei Province (2003A005).
文摘Let W={W(t);t ∈R_+~N} be the d-dimensional N-parameter Brownian Sheet. Sufficient conditions for a compact set F C Rd \ {0} to be a polar set for W are proved. It is also proved that if 2N≤d, then for any compact set E (?)R_>~N ,inf {dimF : F ∈B(Rd), P{W(E) ∩F≠(?)}>0} = d- 2DiroE, and if 2N > d, then for any compact set F C Rd \ {0}, inf{dim E : E ∈ B(R_>~N), P{W(E)∩F≠(?)}>0}=d/2-DimF/2,where B(Rd) and B(R_>~N) denote the Borel σ-algebra in Rd and R_>~N respectively, and dim and Dim are Hausdorff dimension and Packing dimension respectively.
基金Supported by the key research base for humanities and social sciences of Zhejiang Provincial high education talents (Statistics of Zhejiang Gongshang University)
文摘Let W^~ be a two-parameter, R^d-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under W^~. He also presents the connections between the Lebesgue measure of the image of W^~ and Bessel-Riesz capacity. His conclusions also solve a problem proposed by J.- P.Kahane.
基金the National Natural Science Foundation of China (10471148)the Natural Science Foundation of Shaanxi Province (2005A08, 2006A14)
文摘Let W^-(t)(t∈R+^N) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-(t). It is proved that for any α∈ R^d, P{W^-(t) = α, for some t∈ R〉^N} = {1, if βd 〈 2N ,0 if αd〉 2N and the probability that W^-(t) has k-multiple points is 1 or 0 according as whether 2kN〉d(k-1)β or 2kN 〈 d(k - 1)α. These results contain and extend the results of the Brownian sheet, where R〉^N = (0,+∞)U,R+^N = [0,+∞)^N,0〈 α ≤1and β〉1.
基金Supported by the National Natural Science Foundation of China(No.70471071)the Key Research Base for Humanities and Social Sciences of Zhejiang Provincial High Education Talents(Statistics of Zhejiang Gongshang University).
文摘Let B^H,K : (B^H,K(t), t ∈R+^N} be an (N,d)-bifractional Brownian sheet with Hurst indices H = (H1,..., HN) ∈ (0, 1)^N and K = (K1,..., KN)∈ (0, 1]^N. The characteristics of the polar functions for B^H,K are investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of B^H,K is presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov's entropy index for B^H,K are obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.
基金Key Research Base for Humanities and Social Sciences of Zhejiang Provincial High Education Talents(Statistics of Zhejiang Gongshang University)
文摘Let W^~=^~{W^~(t); t∈ R+^N) be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for (t,W^~(t)) are proved. It is also proved that if 2N ≤αd, then for any compact set E ∩→ R〉^N,d-2/α Dim E≤inf{dim F:F∈B(R^d),P{W^~(E)∩F≠0}〉0}≤d-2/βDimE,and if 2N〉αd, then for any compact set F∪→R^d/{0},α/2(d-DimF)≤inf{dimE:E∈B(R〉^N),P{W^~(E)∩F≠0}〉0}≤β/2(d-DimF),where B(R^d) and B(R〉^N) denote the Borel σ-algebra in R^d and in R〉^N respectively, dim and Dim are Hausdorff dimension and Packing dimension respectively.
基金Supported by the National Natural Science Foundation of China
文摘For the N-parameter d-dimensional Generalized Brownian Sheet (in short G. B. S.) satisfying some conditions, this paper proves that when 2N≤αd, with probability one, E∈B(R_+~N), there holds 2/βdim E+2N-2Nβ/α≤dim (E)≤2/αdim E. Especially when α=β=1, we have dim (E)=2 dim E. Noting that even if α=β=1, the G. B. S. is wider than the Brownian Sheet, thus we have extended the uniform dimension result of Mountford, T,S.
基金Supported by the National Natural Science Foundation of China(No.10571159)Specialized Research Found for Doctor Program of Higher Education(No.20060335032)Hangdian Foundation(No.KYS091506042).
文摘Let B0^H = {B0^H(t),t ∈ R+^N) be a real-valued fractional Brownian sheet. Define the (N,d)- Gaussian random field B^H by B^H(t) = (B1^H(t),...,Bd^H(t)) t ∈ R+^N, where B1^H, ..., Bd^H are independent copies of B0^H. The existence and joint continuity of local times of B^H is proven in some given conditions in [22]. We then study further properties of the local times of B^H, such as the moments of increments of local times, the large increments and the maximum moduli of continuity of local times and as a result, we answer the questions posed in [22].
基金The research of Yazhen Wang was supported in part by NSF[grant number DMS-12-65203][grant number DMS-15-28375].
文摘Monte Carlo and quasi-Monte Carlomethods arewidely used in scientific studies.As quasi-Monte Carlo simulations have advantage over ordinaryMonte Carlomethods,this paper proposes a new quasi-Monte Carlo method to simulate Brownian sheet via its Karhunen–Loéve expansion.The proposed new approach allocates quasi-random sequences for the simulation of random components of the Karhunen–Loéve expansion by maximum reducing its variability.We apply the quasi-MonteCarlo approach to an option pricing problem for a class of interest rate models whose instantaneous forward rate driven by a different stochastic shock through Brownian sheet andwe demonstrate the application with an empirical problem.
基金supported by the Natural Science Foundation of Guangxi(Grant No.2020GXNSFAA159118)Guangxi Science and Technology Project(Grant No.Guike AD20297006)the Innovation Project of School of Mathematics and Computing Science of GUET Graduate Education(Nos.2021YJSCX05,2022YJSCX04)。
文摘In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and small deviation for a Brownian sheet.