In page 617 of the paper "Reaction-Diffusion equations for Interacting Particle Systems" from De Masi Ferrari and Lebowitz, one reads:
$ \ $ $ \ $ From classical estimates on the sum of independent random variables (see, for instance, Petrov$^{(44)}$) we have $\forall z\in\mathbb{Z}$ $$\lim_{\varepsilon\to0}\sum_y\left|G_t^\varepsilon(z,y)-\frac{\varepsilon}{\sqrt{\pi t}}\exp\left\{-\frac{(\varepsilon y)^2}{2t}\right\}\right|=0\tag{4.15}$$
Here $G^\varepsilon_t(z,y)$ is the probability that a simple symmetric random walk with jump rate $\varepsilon^{-2}$ starting at $z$ reaches $y$ in $t$ seconds.
Let $X_t$ denote the position of the walk after $t$ seconds. We know that
$$\frac{X_t - z}{t \varepsilon^{-2}} \to N(0,1) $$
I guess that this is somewhat related to (4.15), but I can't prove it.
Any ideas?