Why does this (IID) random variable NOT follow Central Limit Theorem?

Question

We have independent identically distributed random variables $X_i$, which can only take values on selected integers with the following probabilities: $$p(1) = 70/128$$ $$p(9) = 42/128$$ $$p(25) = 14/128$$ $$p(49) = 2/128$$ probability for any other integers are $0$.

Numerical simulations show that the sample average: $${\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}}$$ does NOT follow central limit theorem (there is no sign of converge to a normal distribution), why ?

Answer 1

The CLT guarantees that $\bar{X}$ converges to a normal distribution as $n \rightarrow \infty$, but it makes no guarantees about how quick that convergence will be. Here's what $X_1$ looks like as a line plot:

Here's what $\sum_{i = 1}^n X_i$ looks like when $n = 2$:

It's shifted a tiny bit towards being more symmetrical, but I will admit that it doesn't look very bell-curve-like yet. If we go to $n = 8$, it looks something like this:

Now there is a clear bell-curve appearing. It's still spiky, because there are plenty of values that it can't take, but the behaviour matches what the CLT expects.

If we take it up to $n = 256$, then we have something that is indisputably bell-shaped, although at this scale you can actually see banding of two different bell curves hiding in there:

If we were then to scale this, by taking $\frac{\sum X_i - n E(X)}{\sqrt{n Var(X)}}$, and plot it as a histogram, then you'd get something very close to a standard normal distribution.

Since I've updated the images to fix an error in how I calculated the probabilities, here's the hacked-together R code I used so that others can confirm that I didn't make any other mistakes:

oprobs <- rep(0, 49)
oprobs[1] <- 70/128
oprobs[9] <- 42/128
oprobs[25] <- 14/128
oprobs[49] <- 2/128

probs <- oprobs
probs <- convolve(probs, rev(probs), type = "open") # Repeat this line to double n

plot(probs, type = "l")