When does a pointwise CLT hold?

Question

Let $X$ be a random variable with mean $0$ and variance $1$, and let $X_1, X_2, X_3, \dots$ be iid copies of $X$. Under what conditions can we say that the density of $\frac{X_1+\dots+X_n}{\sqrt{n}}$ converges pointwise to $N(0,1)$?

In particular, when can I say that for any sequence $\epsilon_n \rightarrow 0$ we have $$\frac{P(|\frac{X_1+\dots+X_n}{\sqrt{n}}|<\epsilon_n)-P(|N(0,1)|<\epsilon_n)}{\epsilon_n} \rightarrow 0?$$

In flavor this is somewhat similar to what I've seen termed as "local limit theorems", except a little bit stronger; for example if $X$ is a Bernoulli variable the above would not hold (take $\epsilon_n=2^{-n}$). My guess would be that a sufficient condition would be for the usual CLT to hold and $X$ to have bounded density functions, though I haven't seen this cited anywhere.

Answer 1

Bounded density will suffice, I think. Basically what one needs is for the Fourier transforms (aka characteristic functions) of the $X_1 + \ldots + X_n / \sqrt{n}$ to converge pointwise to the Fourier transform of normal distribution while being dominated by something integrable plus something whose L^1 norm goes to zero, so that the (noisy) Lebesgue dominated convergence theorem applies and will give uniform convergence of the density function. Pointwise convergence is not a problem, because the finite second moment will make the characteristic function of X in the class $C^2$ (twice continuously differentiable). This function cannot equal 1 except at the origin (because X is not discrete), so by continuity and Riemann-Lebesgue it is bounded by $1-\epsilon$ outside of a small neighbourhood of the origin; this together with Plancherel (here we use the bounded density - actually square integrable density will suffice) is enough to get the required domination.

Answer 2

The pointwise (actually, uniform) convergence of densities is indeed the material of classical local limit theorems due to Gnedenko, see his classical book coauthored with Kolmogorov:

Limit distributions for sums of independent random variables, by B. V. Gnedenko and A. N. Kolmogorov. Translated from the Russian and annotated, by K. L. Chung. With an appendix by J. L. Doob. Cambridge, Mass., Addison-Wesley, 1954.

One of possible sufficient conditions is that the density is of bounded variation on ℝ. More generally, one may require that the density is L^p for p∈(1,2] and a certain smoothness property.

Gnedenko also has very nice local theorems for lattice-valued r.v.'s in another chapter of the same book.

Answer 3

Feller states the Berry-Esseen theorem in the following way. Let the $X_k$ be independent variables with a common distribution $F$ such that $$E[X_k]=0, E[X_k^2]=\sigma^2>0, E[|X_k|^3]=\rho<\infty,$$ and let $F_n$ stand for the distribution of the normalized sum $$(X_1+ \dots X_n)/(\sigma \sqrt{n}).$$ Then for all $x$ and $n$ $$|F_n(x)-N(x)| \leq \frac{3\rho}{\sigma^3 \sqrt{n}}.$$

The expression you are interested in is $$\left|\frac{F_n(\epsilon)-F_n(-\epsilon)-N(\epsilon)+N(-\epsilon)}{\epsilon}\right|,$$ which is less than $$\left| \frac{F_n(\epsilon)-N(\epsilon)}{\epsilon} \right| + \left| \frac{F_n(-\epsilon)-N(-\epsilon)}{\epsilon} \right|,$$ which by Berry-Esseen is bounded by $$2\frac{3\rho}{\epsilon \sigma^3 \sqrt{n}}.$$ So, if $\epsilon\sqrt{n}$ goes to infinity, then you are good.

I realize this isn't what you asked, in that you wanted conditions on $X$, and this instead gives you conditions on $\epsilon_n$. Still, perhaps it'll help.

Reference: Feller, An Introduction to Probability Theory and Its Applications, Volume II, Chapter XVI.5.