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Random matrix approximation of spectra of integral operators. (English) Zbl 0949.60078

Summary: Let \(H:L_2(S, {\mathcal S},P) \mapsto L_2(S,{\mathcal S},P)\) be a compact integral operator with a symmetric kernel \(h\). Let \(X_i\), \(i\in\mathbb{N}\), be independent \(S\)-valued random variables with common probability law \(P\). Consider the \(n\times n\) matrix \(\widetilde H_n\) with entries \(n^{-1}h(X_k, X_j)\), \(1\leq i,j\leq n\) (this is the matrix of an empirical version of the operator \(H\) with \(P\) replaced by the empirical measure \(P_n)\), and let \(H_n\) denote the modification of \(\widetilde H_n\), obtained by deleting its diagonal. It is proved that the \(\ell_2\) distance between the ordered spectrum of \(H_n\) 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 \(H_n\) (or \(\widetilde H_n)\) 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).

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
47G10 Integral operators
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