Spectral convergence of empirical integral operators with discontinuous kernels

Abstract

We study the spectral behavior as the sample size $n\to +\infty$ of integral operators defined by convolution of a non-negative symmetric kernel k with respect to empirical measures ${\mu }_n=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{X_i}$, where $\{X_i{\}}_{i=1}^n$ are independent uniform samples from a compact probability metric space $(\mathcal{X},d,\mu )$. Relaxing the usual positivity and continuity assumptions on k, we prove the convergence of these empirical operators to their continuous counterparts, and provide explicit convergence rates.