Precise Large Deviation for the Difference of Non-Random 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).
 Author Name Affiliation Zhiqiang HUA College of Mathematics, Inner Mongolia University for the Nationalities, Inner Mongolia 028043, P. R. China Lixin SONG School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
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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 $$\lim_{t\rightarrow\infty}\sup_{x\geq\gamma (n_{1}(t))^{p+1}}\Big|\frac{P(\sum_{j=1}^{n_1(t)}X_{1j}-\sum_{j=1}^{n_2(t)}X_{2j}-(\mu_{1}n_{1}(t)-\mu_{2}n_{2}(t))>x)}{n_{1}(t)\bar{F_{1}}(x)}-1\Big|=0.$$