Random matrix approximation of spectra of integral operators

Abstract

Let$H:L_2(S,calS,P)\to L_2(S,calS,P)$ be a compact integral operator with a symmetric kernel h. Let$X_i,i\in N$ , be independent S-valued random variables with common probability law P. Consider the n×n matrix${\tilde{H}}_n$ with entries$n^{-1}h(X_i,X_j),1\le i,j\le n$ (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${\tilde{H}}_n,$ 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${\tilde{H}}_n$ ) and H are also obtained under somewhat stronger conditions. These results apply in particular to the kernels of certain functions$H=varphi\left(L\right)$ of partial differential operators L (heat kernels, Green functions).