1-相依假设下最大点间距离的极限定律

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

中文关键词:  最大点间距离  极值分布  Chen-Stein泊松逼近  中偏差  1-相依

英文关键词:maximum interpoint distance  extreme-value distribution  Chen-Stein Poisson approximation  moderation deviation  1-dependent

中文摘要:

      设$\mathcal{M}_{n,p}=( X_{i,k})_{n \times p}$是一个$n\times p$随机矩阵，其$p$列$\boldsymbol{X}^{(1)},\ldots,\boldsymbol{X}^{(p)}$是来自$n$维1-相依高斯总体的独立同分布随机样本.我们不再研究$p$和$n$可比的特殊情形,而是考虑一个更一般的情形,其中$\log_{}{n}=o ( p^{1/3})$.我们证明了最大点间距离$M_{n}=\max\{\vert \boldsymbol{X}_{i}-\boldsymbol{X}_{j}\vert; 1\le i< j\le n\}$收敛到一个极值分布,其中$\boldsymbol{X}_{i}$和$\boldsymbol{X}_{j}$分别表示$\mathcal{M}_{n,p}$的第$i$行和第$j$行.证明过程利用了Chen-Stein泊松逼近方法和中等偏差原理.

英文摘要:

      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.