Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator

Question

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.

Answer 1

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, 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.