# The limits of central limit theorem – part 2

In the previous part we made look through the distribution of sample means for three distributions: Uniform, Cauchy, and Petersburg distribution. The Cauchy and Petersburg distributions do not fulfill the Central Limit Theorem since they have infinite variance (and infinite expected value in “Petersburg” case). Now we will have a look at the numerical results for standard deviation of sample means. As in previous part, we use Uniform distribution only as a reference since it fulfills CLT and we use the same pseud-random number generator (Mersenne-Twister).

Uniform distribution, gives a really nice impressions when looking through the chart showing standard deviation $$\sigma$$ vs sample size $$n$$:

It behaves as we expected – in accordance to the CLT – it varies like: $$\frac{1}{\sqrt{n}}$$.

And now – stars of our show – Cauchy distribution:

In spite of trying many different pseudo-random generator seed values the result is the roughly the same – $$\sigma(n)$$ does not seem to converge.