Random matrix approximation of spectra of integral operators

Abstract

Let$H:L_2(S,{\cal S},P) \rightarrow L_2(S,{\cal S},P)$ 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${n^{-1}h(X_i, X_j),\ 1\leq i,j\leq n}$ (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${\tilde H_n,}$ 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${H=\varphi (L)}$ of partial differential operators L (heat kernels, Green functions).