Limit Laws for the Maximum Interpoint Distance under a 1-Dependent Assumption

Abstract:

      Let $\mathcal{M} _{n,p}=(X_{i,k})_{n \times p}$ be an $n \times p$ random matrix whose $p$ columns $\boldsymbol{X}^{(1)},\ldots,\boldsymbol{X}^{(p)}$ are an $n$-dimensional i.i.d. random sample of size $p$ from 1-dependent Gaussian populations. Instead of investigating the special case where $p$ and $n$ are comparable, we consider a much more general case in which $\log_{}{n}=o(p^{1/3})$. We prove that the maximum interpoint distance $M_{n}=\max\{\vert \boldsymbol{X}_{i}-\boldsymbol{X}_{j}\vert; 1\le i< j\le n\}$ converges to an extreme-value distribution, where $\boldsymbol{X}_{i}$ and $\boldsymbol{X}_{j}$ denote the $i$-th and $j$-th row of $\mathcal{M}_{n,p}$, respectively. The proofs are completed by using the Chen-Stein Poisson approximation method and the moderation deviation principle.