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Jack W Silverstein - One of the best experts on this subject based on the ideXlab platform.
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no eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix
Journal of Multivariate Analysis, 2009Co-Authors: Debashis Paul, Jack W SilversteinAbstract:We consider a class of matrices of the form C"n=(1/N)A"n^1^/^2X"nB"nX"n^*xA"n^1^/^2, where X"n is an nxN matrix consisting of i.i.d. standardized complex entries, A"n^1^/^2 is a Nonnegative Definite square root of the Nonnegative Definite Hermitian matrix A"n, and B"n is diagonal with Nonnegative diagonal entries. Under the assumption that the distributions of the eigenvalues of A"n and B"n converge to proper probability distributions as nN->[email protected]?(0,~), the empirical spectral distribution of C"n converges a.s. to a non-random limit. We show that, under appropriate conditions on the eigenvalues of A"n and B"n, with probability 1, there will be no eigenvalues in any closed interval outside the support of the limiting distribution, for sufficiently large n. The problem is motivated by applications in spatio-temporal statistics and wireless communications.
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clt for linear spectral statistics of large dimensional sample covariance matrices
Annals of Probability, 2004Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let Bn=(1/N)T1/2nXnX∗nT1/2n where Xn=(Xij) is n×N with i.i.d. complex standardized entries having finite fourth moment, and T1/2n is a Hermitian square root of the Nonnegative Definite Hermitian matrix Tn. The limiting behavior, as n→∞ with n/N approaching a positive constant, of functionals of the eigenvalues of Bn, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of Bn, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be 1/n by proving, after proper scaling, that they form a tight sequence. Moreover, if \exppX211=0 and \expp|X11|4=2, or if X11 and Tn are real and \exppX411=3, they are shown to have Gaussian limits.
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exact separation of eigenvalues of large dimensional sample covariance matrices
Annals of Probability, 1999Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let $B _n = (1/N) T_n^{1/2} X _n X _n^*T_n^{1/2}$ where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the Nonnegative Definite Hermitian matrix $T_n$. It was shown in an earlier paper by the authors that, under certain conditions on the eigenvalues of $T_n$, with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large $n$. For these $n$ the interval corresponds to one that separates the eigenvalues of $T_n$. The aim of the present paper is to prove exact separation of eigenvalues; that is, with probability 1, the number of eigenvalues of $B_n$ and $T_n$ lying on one side of their respective intervals are identical for all large $n$.
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no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices
Annals of Probability, 1998Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let $B_n = (1/N)T_n^{1/2}X_n X_n^* T_n^{1/2}$, where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment and $T_n^{1/2}$ is a Hermitian square root of the Nonnegative Definite Hermitian matrix $T_n$. It is known that, as $n \to \infty$, if $n/N$ converges to a positive number and the empirical distribution of the eigenvalues of $T_n$ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of $T_n$, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all $n$ sufficiently large.
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no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices
Annals of Probability, 1998Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let B n = (1/N)T n 1/2 X n X n *Tn 1/2 , where X n is n x N with i.i.d. complex standardized entries having finite fourth moment and T n 1/2 is a Hermitian square root of the Nonnegative Definite Hermitian matrix T n . It is known that, as n → ∞, if n/N converges to a positive number and the empirical distribution of the eigenvalues of T n converges to a proper probability distribution, then the empirical distribution of the eigenvalues of B n converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of T n , for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all n sufficiently large.
Xian Zhang - One of the best experts on this subject based on the ideXlab platform.
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The rank-constrained Hermitian Nonnegative-Definite and positive-Definite solutions to the matrix equation
2014Co-Authors: Xian ZhangAbstract:We propose necessary and sufficient conditions for the existence of Hermitian Nonnegative-Definite or positive-Definite solutions to the matrix equation AXB = C. A representation of these solutions is also given.
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THE GENERAL HERMITIAN Nonnegative-Definite SOLUTION TO THE MATRIX EQUATION AXA ∗ +BY B ∗ = C
2014Co-Authors: Acta Mathematica, Xian ZhangAbstract:Abstract. Consider the matrix equation AXA ∗ +BY B ∗ = C. A matrix pair (X0, Y0) is called a Hermitian Nonnegative-Definite solution to the matrix equation if X0 and Y0 are Hermitian Nonnegative-Definite and sat-isfy AX0A ∗ + BY0B ∗ = C. We give necessary and sufficient conditions for the existence of a Hermitian Nonnegative-Definite solution to the matrix equa-tion, and further derive a representation of the general Hermitian Nonnegative-Definite solution to the equation when it has such solutions. An example shows these advantages of the proposed approach. 1
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the general common hermitian Nonnegative Definite solution to the matrix equations axa bb and cxc dd with applications in statistics
Journal of Multivariate Analysis, 2005Co-Authors: Xian ZhangAbstract:We deduce a necessary and sufficient condition for the matrix equations AXA* = BB* and CXC* = DD* to have a common Hermitian Nonnegative-Definite solution, and a representation of the general common Hermitian Nonnegative-Definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).
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the general common Nonnegative Definite and positive Definite solutions to the matrix equations axa bb and cxc dd
Applied Mathematics Letters, 2004Co-Authors: Xian Zhang, Meiyu ChengAbstract:Abstract We give necessary and sufficient conditions for the existence of a common Nonnegative-Definite (positive-Definite) solution to the pair of matrix equations AXA∗ = BB∗ and CXC∗ = DD∗ , and derive a representation of the general common Nonnegative-Definite (positive-Definite) solution to these two equations when they have such common solutions. This paper can be viewed as a supplementary version of that derived by Young et al. [1] since Groβ [2] has given a counterexample to point out a mistake in their basic Theorem 1. The presented example shows the advantage of the proposed approach.
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hermitian Nonnegative Definite and positive Definite solutions of the matrix equation axb c
Applied Mathematics E-Notes [electronic only], 2004Co-Authors: Xian ZhangAbstract:We propose necessary and sufficient conditions for the existence of Hermitian Nonnegative-Definite or positive-Definite solutions to the matrix equation AXB = C. A representation of these solutions is also given.
Robert J Elliott - One of the best experts on this subject based on the ideXlab platform.
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discrete time mean field stochastic linear quadratic optimal control problems ii infinite horizon case
Automatica, 2015Co-Authors: Robert J ElliottAbstract:This paper first presents results on the equivalence of several notions of L2L2-stability for linear mean-field stochastic difference equations with random initial value. Then, it is shown that the optimal control of a mean-field linear–quadratic optimal control with an infinite time horizon uniquely exists, and the optimal control can be expressed as a linear state feedback involving the state and its mean, via the minimal Nonnegative Definite solution of two coupled algebraic Riccati equations. As a byproduct, the open-loop L2L2-stabilizability is proved to be equivalent to the closed-loop L2L2-stabilizability. Moreover, the minimal Nonnegative Definite solution, the maximal solution, the stabilizing solution of the algebraic Riccati equations and their relations are carefully investigated. Specifically, it is shown that the maximal solution is employed to construct the optimal control and value function to another infinite time horizon mean-field linear–quadratic optimal control. In addition, the maximal solution being the stabilizing solution, is completely characterized by properties of the coefficients of the controlled system. This enriches the existing theory about stochastic algebraic Riccati equations. Finally, the notion of exact detectability is introduced with its equivalent characterization of stochastic versions of the Popov–Belevitch–Hautus criteria. It is then shown that the minimal Nonnegative Definite solution is the stabilizing solution if and only if the uncontrolled system is exactly detectable.
Zhidong Bai - One of the best experts on this subject based on the ideXlab platform.
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Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices
2016Co-Authors: Zhidong Bai, Li Huiqin, Guangming PanAbstract:Suppose that $\mathbf X_n=(x_{jk})$ is $N\times n$ whose elements are independent real variables with mean zero, variance 1 and the fourth moment equal to three. The separable sample covariance matrix is defined as $\mathbf{B}_n = \frac1N\mathbf{T}_{2n}^{1/2} \mathbf{X}_n \mathbf{T}_{1n} \mathbf{X}_n' \mathbf{T}_{2n}^{1/2}$ where $\mathbf{T}_{1n}$ is a symmetric matrix and $\mathbf{T}_{2n}^{1/2}$ is a symmetric square root of the Nonnegative Definite symmetric matrix $\mathbf{T}_{2n}$. Its linear spectral statistics (LSS) are shown to have Gaussian limits when $n/N$ approaches a positive constant
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clt for linear spectral statistics of large dimensional sample covariance matrices
Annals of Probability, 2004Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let Bn=(1/N)T1/2nXnX∗nT1/2n where Xn=(Xij) is n×N with i.i.d. complex standardized entries having finite fourth moment, and T1/2n is a Hermitian square root of the Nonnegative Definite Hermitian matrix Tn. The limiting behavior, as n→∞ with n/N approaching a positive constant, of functionals of the eigenvalues of Bn, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of Bn, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be 1/n by proving, after proper scaling, that they form a tight sequence. Moreover, if \exppX211=0 and \expp|X11|4=2, or if X11 and Tn are real and \exppX411=3, they are shown to have Gaussian limits.
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exact separation of eigenvalues of large dimensional sample covariance matrices
Annals of Probability, 1999Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let $B _n = (1/N) T_n^{1/2} X _n X _n^*T_n^{1/2}$ where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the Nonnegative Definite Hermitian matrix $T_n$. It was shown in an earlier paper by the authors that, under certain conditions on the eigenvalues of $T_n$, with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large $n$. For these $n$ the interval corresponds to one that separates the eigenvalues of $T_n$. The aim of the present paper is to prove exact separation of eigenvalues; that is, with probability 1, the number of eigenvalues of $B_n$ and $T_n$ lying on one side of their respective intervals are identical for all large $n$.
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no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices
Annals of Probability, 1998Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let $B_n = (1/N)T_n^{1/2}X_n X_n^* T_n^{1/2}$, where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment and $T_n^{1/2}$ is a Hermitian square root of the Nonnegative Definite Hermitian matrix $T_n$. It is known that, as $n \to \infty$, if $n/N$ converges to a positive number and the empirical distribution of the eigenvalues of $T_n$ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of $T_n$, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all $n$ sufficiently large.
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no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices
Annals of Probability, 1998Co-Authors: Zhidong Bai, Jack W SilversteinAbstract:Let B n = (1/N)T n 1/2 X n X n *Tn 1/2 , where X n is n x N with i.i.d. complex standardized entries having finite fourth moment and T n 1/2 is a Hermitian square root of the Nonnegative Definite Hermitian matrix T n . It is known that, as n → ∞, if n/N converges to a positive number and the empirical distribution of the eigenvalues of T n converges to a proper probability distribution, then the empirical distribution of the eigenvalues of B n converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of T n , for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all n sufficiently large.
Le Chen - One of the best experts on this subject based on the ideXlab platform.
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NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY
'Springer Science and Business Media LLC', 2019Co-Authors: Le Chen, Kim KunwooAbstract:In this article, we study the nonlinear stochastic heat equation in the spatial domain (d) subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, Nonnegative and Nonnegative-Definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z(d) to that on (d). Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.11Nsciescopu
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parabolic anderson model with space time homogeneous gaussian noise and rough initial condition
Journal of Theoretical Probability, 2018Co-Authors: Raluca M Balan, Le ChenAbstract:In this article, we study the parabolic Anderson model driven by a space-time homogeneous Gaussian noise on \(\mathbb {R}_{+} \times \mathbb {R}^d\), whose covariance kernels in space and time are locally integrable Nonnegative functions which are Nonnegative Definite (in the sense of distributions). We assume that the initial condition is given by a signed Borel measure on \(\mathbb {R}^d\), and the spectral measure of the noise satisfies Dalang’s (Electron J Probab 4(6):29, 1999) condition. Under these conditions, we prove that this equation has a unique solution, and we investigate the magnitude of the p-th moments of the solution, for any \(p \ge 2\). In addition, we show that this solution has a Holder continuous modification with the same regularity and under the same condition as in the case of the white noise in time, regardless of the temporal covariance function of the noise.
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Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency
2015Co-Authors: Le Chen, Kim KunwooAbstract:In this paper, we study the stochastic heat equation in the spatial domain $\mathbb{R}^d$ subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, Nonnegative and Nonnegative-Definite function that satisfies {\it Dalang's condition}. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. As a first application of these moments bounds, we find the necessary and sufficient conditions for the solution to have phase transition for the second moment Lyapunov exponents. As another application, we prove a localization result for the intermittency fronts.Comment: 26 page
