Law of the logarithm for the maximum interpoint distance constructed by high-dimensional random matrix

2023/12/26 by Zhang, Haibin, Zhang, Yong, Ding, Xue
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2312.15857

Abstract

Suppose \ X_i,k; 1≤ i ≤ p, 1≤ k ≤ n \ is an array of i.i.d.~real random variables. Let \ p=p_n; n ≥1 \ be positive integers. Consider the maximum interpoint distance M_n=max_{1≤ i< j≤ p} ‖ \boldsymbolX_i- \boldsymbolX_j ‖₂ where \boldsymbolX_i and \boldsymbolX_j denote the i-th and j-th rows of the p × n matrix M _p,n=( X_i,k )_{p × n}, respectively. This paper shows the laws of the logarithm for M_n under two high-dimensional settings: the polynomial rate and the exponential rate. The proofs rely on the moderation deviation principle of the partial sum of i.i.d.~random variables, the Chen--Stein Poisson approximation method and Gaussian approximation.

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