Precise Large Deviation for the Difference of Non-Random Sums of NA Random Variables

DOI：10.3770/j.issn:2095-2651.2016.06.013

 作者 单位 华志强 内蒙古民族大学数学学院, 内蒙古 通辽 028043 宋立新 大连理工大学数学科学学院, 辽宁 大连 116024

研究形如$\sum_{j=1}^{n_1(t)}X_{1j}-\sum_{j=1}^{n_2(t)}X_{2j}$的非随机和的差的精确大偏差, 其中$\{X_{1j},j\geq 1\}$是一列服从共同分布$F_{1}(x)$ 的负相协随机变量序列, $\sum_{j=1}^{n_1(t)}X_{1j}$是$\{X_{1j},j\geq 1\}$的非随机和, $\{X_{2j},j\geq 1\}$是一列服从独立同分布的随机变量序列, $\sum_{j=1}^{n_2(t)}X_{2j}$是$\{X_{2j},j\geq 1\}$的非随机和, $n_1(t)$和$n_2(t)$是两个取正整数的函数. 在一些其它的条件下,得到了如下一致渐近关系$$\lim_{t\rightarrow\infty}\sup_{x\geq\gamma (n_{1}(t))^{p+1}}|\frac{P(\sum_{j=1}^{n_1(t)}X_{1j}-\sum_{j=1}^{n_2(t)}X_{2j}-(\mu_{1}n_{1}(t)-\mu_{2}n_{2}(t))>x)}{ n_{1}(t)\bar{F_{1}}(x)}-1|=0.$$

In this paper, we study precise large deviation for the non-random difference $\sum_{j=1}^{n_1(t)}X_{1j}$ $-\sum_{j=1}^{n_2(t)}X_{2j}$, where $\sum_{j=1}^{n_1(t)}X_{1j}$ is the non-random sum of $\{X_{1j},j\geq 1\}$ which is a sequence of negatively associated random variables with common distribution $F_{1}(x)$, and $\sum_{j=1}^{n_2(t)}X_{2j}$ is the non-random sum of $\{X_{2j},j\geq 1\}$ which is a sequence of independent and identically distributed random variables, $n_1(t)$ and $n_2(t)$ are two positive integer functions. Under some other mild conditions, we establish the following uniformly asymptotic relation $$\lim_{t\rightarrow\infty}\sup_{x\geq\gamma (n_{1}(t))^{p+1}}\Big|\frac{P(\sum_{j=1}^{n_1(t)}X_{1j}-\sum_{j=1}^{n_2(t)}X_{2j}-(\mu_{1}n_{1}(t)-\mu_{2}n_{2}(t))>x)}{n_{1}(t)\bar{F_{1}}(x)}-1\Big|=0.$$