# Central Limit Theorem on the Circle

I am interested in a circular equivalent to the classical CLT. Is there a necessary and sufficient condition telling when a normalized sum of circular distributed random variables converges to a WrappedNormal distributed random variable? That is, what conditions do circular i.i.d. random vectors $X_i$ have to satisfy in order to ensure $$\frac{1}{\sqrt{n}} \sum_{i=1}^n X_i \mod 2\pi$$ converges to a WrappedNormal distributed random variable. Is there a similar result for (hyper-)spheres?

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Please define "normalized sum of circular distributed random variables". – Did Aug 14 '13 at 20:51
I mean a similar summation scheme as in the case of the classical CLT. Added it to the original post. – Igor Aug 17 '13 at 20:29
How do you define the division on the circle? – D. Thomine Aug 17 '13 at 20:33
It seems that 5 upvoters know the answer to the query formulated twice in the comments above. – Did Aug 17 '13 at 22:33
@D.Thomine: Not at all. In order to account for the fact, that $X_i$ is circular, we can simply use a real-valued random vector restricted to $[0, 2\pi)$. Thus, we use the classical division for real numbers. The summation scheme presented in my post obviously yields again a random vector restricted to $[0, 2\pi)$. – Igor Aug 19 '13 at 11:30