Philosophical question about central limit theorem

[coquelicot]

Well, this is probably a stupid question, but I don't see why (yet).

Let X_i be random variables identically distributed, with mean 0, such that the cumulative distribution is = 0 for all -1 < x < 1. So, I believe it is clear that for all n, the cumulative distribution of Z = (X₁ + X₂ ... X_n)/n is = 0 for all x < -1. But the central limit theorem implies that this distribution (normalized by the square root of n), converges in distribution to the Normal distribution. So, for some n sufficiently large, the cumulative distribution of Z must be > 0 for some x < -1. Where is the fallacy in this paradox?
thx.

 

[Stephen Tashi]

But the central limit theorem implies that this distribution (normalized by the square root of n),
.

The normalization involves multiplying the sample mean by $\sqrt{n}$. The quantity that is approximately normally distributed (for a sample of size n , sample mean $S_n$ and population mean $\mu$ for each individual random variable) is $\sqrt{n}(S_n - \mu)$.

 

[chiro]

If you have issues with the application of the theorem to a particular distribution I would recommend opening a statistical software package and simulating a number of random variables and then graphing the result of the random vectors that get the arithmetic mean and see what happens.

It should help convince you through some examples of how the CLT holds.

 

[coquelicot]

Thx all.
I have understood my error.