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| Local Precise Large and Moderate Deviations for Sums of Independent Random Variables |
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Citation： |
Fengyang CHENG,Minghua LI.Local Precise Large and Moderate Deviations for Sums of Independent Random Variables[J].Chinese Annals of Mathematics B,2016,37(5):753~766 |
| Page view： 1198
Net amount： 1006 |
Authors： |
Fengyang CHENG; Minghua LI |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11401415). |
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| Abstract： |
Let $\{X,X_k: k\geq1\}$ be a sequence of independent and identically
distributed random variables with a common distribution $F$. In this
paper, the authors establish some results on the local precise large
and moderate deviation probabilities for partial sums
$S_n=\sum\limits_{i=1}^nX_i$ in a unified form in which $X$ may be a
random variable of an arbitrary type, which state that under some
suitable conditions, for some constants $T>0,\ a$ and $\tau>\frac12$
and for every fixed $\gamma>0$, the relation
\begin{align*}
P(S_n-na\in (x,x+T])\sim n F((x+a,x+a+T])
\end{align*}
holds uniformly for all $x\geq \gamma n^{\tau}$ as $n\to\infty$,
that is,
\begin{align*}
\lim_{n\to+\infty}\sup_{x\geq \gamma n^\tau}\Big|\frac{P(S_n-na\in
(x,x+T])}{n F((x+a,x+a+T])}-1\Big|=0.
\end{align*}
The authors also discuss the case where $X$ has an infinite mean. |
Keywords： |
Local precise moderate deviation, Local precise large deviation,
Intermediate regularly varying function, $O$-regularly varying
function |
Classification： |
60F10 |
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