Local Precise Large and Moderate Deviations for Sums of Independent Random Variables

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.