The smallest eigenvalues of random kernel matrices: Asymptotic results on the min kernel

This paper investigates asymptotic properties of the smallest eigenvalue of the random kernel matrix M-n = [1/K(X-i, X-j)](ij=1)(n), where k(x, y) = min{x,y} is the min kernel function and X-1, X-2, ..., X-n are i.i.d. random variables in [0, 1]. We prove that under certain conditions, the smallest eigenvalue converges in L-1 to zero with the rate of convergence O(n(-3)). In addition, if the underlying distribution of X-i's has a bounded density, the distribution of the smallest eigenvalue scaled by n(3) converges to an exponential distribution. Published by Elsevier B.V.