Local central limit theorem far from the center

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 $$\exp\left\{-\frac{x^2}2 - xy\right\}\left[1 + \frac{C(x^2 + y^2)}n + o\left(\frac1n\right)\right].$$

• This will very much depend on the tail asymptotics of the density of $X$. – Iosif Pinelis Jun 24 at 17:23
• We can do explicit calculation when X is Gaussian. So I want to know the case that X is subgaussian: $E [e^{sX}] < e^{c^2s^2/2}$ for any $s$ in $R$. – gregarki khayal Jun 24 at 22:54

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), so that $$r_n(x,y)\to1$$ as $$n\to\infty$$.
• Yes, I realized that it depends on the tail of X. I am typically interested in the case that X is sub-Gaussian: $Ee^{sX} < e^{cs^2}$ for all $s$. – gregarki khayal Jun 24 at 23:02