## 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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