Let X be a linear process having a finite fourth moment. Assume F is a class of square-integrable functions. We consider the empirical spectral distribution function J(n,X) based on X and indexed by F. If F is totally...
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Let X be a linear process having a finite fourth moment. Assume F is a class of square-integrable functions. We consider the empirical spectral distribution function J(n,X) based on X and indexed by F. If F is totally bounded then J(n,X) satisfies a uniform strong law of large numbers. If, in addition, a metric entropy condition holds, then J(n,X) obeys the uniform central limit theorem. (C) 1997 Elsevier Science B.V.
A new form of empiricalspectraldistribution of a Wigner matrix W-n with weights specified by the eigenvectors is defined and it is then shown to converge with probability one to the semicircular law. Moreover, centr...
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A new form of empiricalspectraldistribution of a Wigner matrix W-n with weights specified by the eigenvectors is defined and it is then shown to converge with probability one to the semicircular law. Moreover, central limit theorem for linear spectral statistics defined by the eigenvectors and eigenvalues is also established under some moment conditions, which suggests that the eigenvector matrix of W-n is close to being Haar distributed.
Let {X-ij}, i, j = .... be a double array of i.i.d. complex random variables with EX11 =0, E vertical bar X-11 vertical bar(2) = 1 and E vertical bar X-11 vertical bar(4) < infinity, and let A(n) = 1/N T-n(1/2) X-n...
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Let {X-ij}, i, j = .... be a double array of i.i.d. complex random variables with EX11 =0, E vertical bar X-11 vertical bar(2) = 1 and E vertical bar X-11 vertical bar(4) < infinity, and let A(n) = 1/N T-n(1/2) X-n (XnTn1/2)-T-*, where T-n(1/2) is the square root of a nonnegative definite matrix T-n and X-n is the n x N matrix of the upper-left corner of the double array. The matrix A(n) can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix T-n, or as a multivariate F matrix if T-n is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of A(n), a new form of empiricalspectraldistribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectraldistribution as the empiricalspectraldistribution defined by equal weights. Moreover, if {X-ij} and T-n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of A(n) are proved to have Gaussian limits, which suggests that the eigenvector matrix of A(n) is nearly Haar distributed when T-n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.
We consider the asymptotic behavior of the singular values of a so-called spherical ensemble of random matrices of large dimension. These are matrices of the form XY-1, where X and Y are independent matrices of dimens...
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Let {vij}, i, j = 1, 2, …, be i.i.d, random variables with Ev11 = 0, Ev11^2 = 1 and a1 = （ai1,…, aiM） be random vectors with {aij} being i.i.d, random variables. Define XN =（x1,…, xk） and SN =XNXN^T,where xi=ai...
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Let {vij}, i, j = 1, 2, …, be i.i.d, random variables with Ev11 = 0, Ev11^2 = 1 and a1 = （ai1,…, aiM） be random vectors with {aij} being i.i.d, random variables. Define XN =（x1,…, xk） and SN =XNXN^T,where xi=ai×si and si=1/√N（v1i,…, vN,i）^T. The spectraldistribution of SN is proven to converge, with probability one, to a nonrandom distributionfunction under mild conditions.
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