Let $\{X\_i \}\_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X\_1 = 0$ and $\mathbb{E} X\_1 ^2 < \infty$. Assume that $\{S\_n  \}\_{n \in \mathbb{N}}$ is a non-lattice random walk, where $S\_n = X\_1+...+ X\_n$. I am wondering whether there is a 'local large deviation theorem' running along these lines:


**Theorem.** *Let $0 < a < b$. Then as $n \to \infty$ uniformly for $r \in [an, bn]$*
$$
 \mathbb{P} \{  S\_n \in [r,r+1] \} \sim \int\limits\_{r} ^{r+1} \kappa \_n (x) dx,
$$
*where $\kappa \_n$ is a function given explicitly.*


Similar results are available for $r = o(n)$ in [1], at least for absolutely continuous random variables. This question can also be formulated in terms of the associated convolution operator. Let $a \in L^1$, $a \geq 0$. Define the operator $L$ on some function space by
$$
Lu (x) = \int\limits \_{\mathbb{R}} u(y)a(x-y) dy
$$
and set $u\_n = L ^n \delta \_0$ (or alternatively $u\_n = \frac{1}{2 \varepsilon}L ^n \mathbf{1} \_\varepsilon$, where $\mathbf{1} \_\varepsilon$ is the indicator of $[-\varepsilon, \varepsilon]$ for a small $\varepsilon > 0$). What can we say about
$$
\int\limits \_{r} ^{r+1} u\_n (x)dx.
$$
for $r \in [an, bn]$, possibly under some additional assumptions on $a$?


[1]: Richter, W. (1957). Local limit theorems for large deviations. Theory of Probability & Its Applications, 2(2), 206-220.



Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator
For general classes of bounded pdf's of $X\_1$, including pdf's with exponential-like, super-exponential, and sub-exponential tails, your Theorem follows from the considerations in [Sections 2.1 and 2.2](https://www.sciencedirect.com/science/article/abs/pii/009630039490197X), with
$$\kappa\_n(x)=p\_{S\_n}(r)e^{-s\_0(x-r)},$$
where $p\_{S\_n}$ is the pdf of $S\_n$,
$$s\_0:=\sup\{s\ge0\colon m(s)<r/n\},$$
and
$$m(s):=\frac{EX\_1e^{sX\_1}}{Ee^{sX\_1}}.$$
As can also be seen from those considerations, the asymptotics of $p\_{S\_n}(r)$ will very much depend on how heavy the right tail of the distribution of $X\_1$ is.


