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

Abstract

Suppose $\left \{ X_{i,k}; 1\le i \le p, 1\le k \le n \right \} $ is an array of i.i.d.~real random variables. Let $\left \{ p=p_{n}; n \ge1 \right \} $ be positive integers. Consider the maximum interpoint distance $M_{n}=\max_{1\le i< j\le p} \left \| \boldsymbol{X}_{i}- \boldsymbol{X}_{j} \right \|_{2} $ where $\boldsymbol{X}_{i}$ and $\boldsymbol{X}_{j}$ denote the $i$-th and $j$-th rows of the $p \times n$ matrix $\mathcal{M} _{p,n}=\left( X_{i,k} \right)_{p \times n}$, 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.