Abstract
Let be a compact integral operator with a symmetric kernel h. Let , be independent S-valued random variables with common probability law P. Consider the n×n matrix with entries (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 obtained by deleting its diagonal. It is proved that the 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 ) and H are also obtained under somewhat stronger conditions. These results apply in particular to the kernels of certain functions of partial differential operators L (heat kernels, Green functions).
Citation
Vladimir Koltchinskii. Evarist Giné. "Random matrix approximation of spectra of integral operators." Bernoulli 6 (1) 113 - 167, Feb 2000.
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