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Feb 2000 Random matrix approximation of spectra of integral operators
Vladimir Koltchinskii, Evarist Giné
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Bernoulli 6(1): 113-167 (Feb 2000).

Abstract

Let H :L 2(S,cal S,P)L 2(S,cal S,P) be a compact integral operator with a symmetric kernel h. Let X i ,iN , be independent S-valued random variables with common probability law P. Consider the n×n matrix H ˜ n with entries n - 1 h(X i,X j),1i,jn (this is the matrix of an empirical version of the operator H with P replaced by the empirical measure Pn), and let Hn denote the modification of H˜ n , obtained by deleting its diagonal. It is proved that the 2 distance between the ordered spectrum of Hn and the ordered spectrum of H tends to zero a.s. if and only if H is Hilbert-Schmidt. Rates of convergence and distributional limit theorems for the difference between the ordered spectra of the operators Hn (or H˜ n ) and H are also obtained under somewhat stronger conditions. These results apply in particular to the kernels of certain functions H =varphi(L) of partial differential operators L (heat kernels, Green functions).

Citation

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Vladimir Koltchinskii. Evarist Giné. "Random matrix approximation of spectra of integral operators." Bernoulli 6 (1) 113 - 167, Feb 2000.

Information

Published: Feb 2000
First available in Project Euclid: 22 April 2004

zbMATH: 0949.60078
MathSciNet: MR2001E:47080

Keywords: Eigenvalues , Heat kernels , Integral‎ ‎Operators , limit theorems , random matrices

Rights: Copyright © 2000 Bernoulli Society for Mathematical Statistics and Probability

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Vol.6 • No. 1 • Feb 2000
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