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

Suppose $\{X_{i,k};1\le i\le p,1\le k\le n\}$ is an array of i.i.d.~real random variables. Let $\{p=p_n;n\ge 1\}$ be positive integers. Consider the maximum interpoint distance $M_n={\max }_{1\le i<j\le p}{\parallel {\mathit{X}}_i-{\mathit{X}}_j\parallel }_2$ where ${\mathit{X}}_i$ and ${\mathit{X}}_j$ 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.