# Sum of Random Variables with no CLT

Suppose $$Y_n$$ are i.i.d. random variables with $$E[Y_i]=0$$, $$\operatorname{Var}(Y_i)=C>0$$, and denote $$X_n=\frac{1}{n}Y_n$$. Does $$\sum_{i=1}^n X_i$$ converge to $$N(0,\sigma^2)$$ ($$\sigma^2=\frac{\pi}{6}\cdot \operatorname{Var}(Y_i)$$) or is it possible for it to converge to something else? (From Kolmogorovs three-series theorem, the sum converges almost surely).

Is there a simple counter example?

• I believe the variance of the sum should be $\sigma^2=\frac{\pi^2}6\cdot \operatorname{Var}(Y_i)$.
– robjohn
Oct 24, 2018 at 9:31

It can converge to all kinds of distributions. If the common distribution is uniform on $$(-1,1)$$ then the characteristic function is $$\phi (t)=\frac {sin (t)} t$$. The characteristic function of $$\sum X_n$$ is $$\prod \frac {sin (t/n)} {t/n}$$ which has zeros. Since the characteristic function of an an infinitely divisible distribution never vanishes it follows that the the sum has a distribution that is not even infinitely divisible, so certainly not normal.