Let $X\_i$ be a sequence of iid random variables, $E [X] = 0$, $E [X^2] = 1$ and $E [|X|^k] < \infty$ for some $k \ge 3$. 
Classical local CLT says that the density function $f\_n$ of $\frac1{\sqrt n}\sum\_1^n X\_i$ satisfies that 
$$
f\_n(x) - \phi(x)\left(1 + \sum\_{j=1}^{k-2} n^{-\frac j2}P\_j(x)\right) = o\left(n^{-\frac{k-2}2}\right), \quad \phi(x) = \frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}
$$
where $P\_j$ is some $(j + 2)$-order polynomial, and the RHS is uniformly small for $x \in R$. 


This gives us very good estimate for constant $x$. 
My question is that can we get a similar expansion equation for $f\_n(\sqrt nx)$? 
Since $\phi(\sqrt nx)$ decays faster than any polynomial order in $n$, we can not apply the local CLT directly. 


Remark: I consider this in order to estimating the following expression for $x \not= 0$ and $y$: 
$$
\frac1{f\_n(\sqrt nx)}f\_{n-1}\left(\frac{nx + y}{\sqrt{n-1}}\right), \quad n \text{ sufficiently large}. 
$$
When $x = 0$ by local CLT this is bounded by $1 + C(1+y^2)/n + o(1/n)$. 
If $x \not= 0$, I expect the upper bound 
$$
\exp\left\{-\frac{x^2}2 - xy\right\}\left[1 + \frac{C(x^2 + y^2)}n + o\left(\frac1n\right)\right]. 
$$



Local central limit theorem far from the center
The asymptotics of the ratio 
$$r\_n(x,y):=\frac1{f\_n(\sqrt nx)}f\_{n-1}\left(\frac{nx + y}{\sqrt{n-1}}\right)
$$ 
will very much depend on the tail asymptotics of the density (say $f$) of $X$. 


E.g., if $X$ is standard normal, then $r\_n(x,y)\sim\exp\{-x^2/2-xy\}$ as $n\to\infty$. 


If e.g. $f(x)\sim x^{-p}$ for some real $p>0$ as $x\to\infty$, then $f\_n(\sqrt n x)\sim nf(\sqrt n x)$ for each real $x>0$ as $n\to\infty$ (cf. e.g. [Vinogradov](https://books.google.com/books?id=EZMOkwMlES0C&pg=PA212&lpg=PA212&dq=%22On%20asymptotic%20expansions%20in%20limit%20theorems%20on%20large%20deviations%20for%20sums%20of%20independent%20random%20variables%20in%20the%20case%20of%20power%20tails%22&source=bl&ots=If3QnBVODR&sig=ACfU3U1LOOFyyl1bnpshr8ZgMa9GGDa2Aw&hl=en&sa=X&ved=2ahUKEwj01Lz88YLjAhXCGc0KHZlYAoMQ6AEwAHoECAAQAQ#v=onepage&q=%22On%20asymptotic%20expansions%20in%20limit%20theorems%20on%20large%20deviations%20for%20sums%20of%20independent%20random%20variables%20in%20the%20case%20of%20power%20tails%22&f=false)), so that $r\_n(x,y)\to1$ as $n\to\infty$. 


