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

Lu-Jing Huang, Yin-Ting Liao, Lo-Bin Chang and Chii-Ruey Hwang

Statistics & Probability Letters, 2019, vol. 148, issue C, 23-29

Abstract: This 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.

Keywords: Kernel matrix; Min kernel; Smallest eigenvalues; Spacing (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1016/j.spl.2018.12.008

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