Gaussian local limit and large-deviation asymptotics for Lévy measures with a nontrivial absolutely continuous part

Let $\mu$ be a measure on $(0,1]$ such that

$$\int x\,d\mu<\infty.$$

Allow $\mu$ to have atoms and a continuous singular part, and suppose its absolutely continuous part is $g(x)\,dx$, with some $0\leq a<b\leq 1$ and $\varepsilon>0$ satisfying $g(x)\geq\varepsilon$ for all $x\in(a,b)$. Let $T_\beta$ be the tilted sum of arrivals, and let $Y$ be its standardized version. Gaussian local-limit and large-deviation conjecture. The density of $Y$ satisfies

$$f_Y(y)\to\frac{1}{\sqrt{2\pi}}e^{-y^2/2}$$

for every $y$, with no assertion about a rate of convergence. Consequently, if $T$ has the possibly defective density $f$, then

$$f(u)=\sqrt{\frac{\beta'(u)}{2\pi}}e^{C(\beta)-u\beta}\left(1+o(1)\right),$$

where

$$C(\beta)=\int_0^1\left(e^{\beta x}-1\right)\mu(dx).$$

The conjecture would substantially weaken the regularity assumptions imposed on the density in the paper, permitting atoms and a continuous singular component; the claimed asymptotics are asserted without a convergence-rate estimate.