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

Publication date: Available online 4 January 2019Source: Statistics & Probability LettersAuthor(s): Lu-Jing Huang, Yin-Ting Liao, Lo-Bin Chang, Chii-Ruey HwangAbstractThis paper investigates asymptotic properties of the smallest eigenvalue of the random kernel matrix Mn=1nk(Xi,Xj)i,j=1n, where k(x,y)=min{x,y} is the min kernel function and X1,X2,…,Xn are i.i.d. random variables in [0,1]. We prove that under certain conditions, the smallest eigenvalue converges in L1 to zero with the rate of convergence O(n−3). In addition, if the underlying distribution of Xi’s has a bounded density, the distribution of the smallest eigenvalue scaled by n3 converges to an exponential distribution.

Source: Statistics and Probability Letters - January 4, 2019 Category: Statistics Source Type: research