The Spectrum of Random Kernel Matrices: Universality Results for Rough and Varying Kernels

We consider random matrices whose entries are [Formula: see text] or f(‖X i – X j ‖ 2 ) for iid vectors X i ∈ ℝ p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of X i 's is sufficiently nice, El Karoui [The spectrum of Kernel random matrices, Ann. Statist.38(1) (2010) 1–50, MR 2589315 (2011a.62187)] showed that the spectral distributions of these matrices behave as if f is linear in the Marčhenko–Pastur limit. When X i 's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng–Singer [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]]. Two results are shown in this paper: first, it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and second, it is shown that the Gaussian assumptions in Cheng–Singer's result can be removed, answering a question posed in [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]] about the universality of the limiting spectral distribution.