Precise Large Deviation for the Difference of NonRandom Sums of NA Random Variables 
Received:August 21, 2015 Revised:December 01, 2015 
Key Word:
precise large deviation negative association consistently varying tail difference

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11371077; 11571058), the Fundamental Research Funds for the Central Universities (Grant No.DUT15LK19) and the Natural Science Foundation of Inner Mongolia University for the Nationalities (Grant Nos.NMDYB1436; NMDYB1437). 

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Abstract: 
In this paper, we study precise large deviation for the nonrandom 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 nonrandom 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 nonrandom 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 $$\lim_{t\rightarrow\infty}\sup_{x\geq\gamma (n_{1}(t))^{p+1}}\Big\frac{P(\sum_{j=1}^{n_1(t)}X_{1j}\sum_{j=1}^{n_2(t)}X_{2j}(\mu_{1}n_{1}(t)\mu_{2}n_{2}(t))>x)}{n_{1}(t)\bar{F_{1}}(x)}1\Big=0.$$ 
Citation: 
DOI:10.3770/j.issn:20952651.2016.06.013 
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