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]. $ $