Convergence of distribution

Question

This is from Probability and Measure by Billingsley, 3rd Edition.

27.21 (p. 370) Let $X_1, X_2,...$ be independent and identically distributed, and suppose that the distribution common to the $X_n$ is supported by $[0,2\pi]$ and is not a lattice distribution. Let $S_n=X_1+\cdots+X_n$, where the sum is reduced modulo $2\pi$. Show that $S_n \Rightarrow U$, where $U$ is uniformly distributed over $[0,2\pi]$.

Can someone provide some hints? Thanks!

P.S. This problem refers back to other two problems. Namely:

26.1 (p. 353) A random variable has a lattice distribution if for some $a$ and $b$, $b>0$, the lattice $\{a+nb:n=0,\pm 1,\dots\}$ supports the distribution of $X$. Let $X$ have characteristic function $\varphi$. (a) Show that a necessary condition for $X$ to have a lattice distribution is that $|\varphi(t)|=1$ for some $t\neq 0$. (b) Show that the condition is sufficient as well. (c) Suppose that $|\varphi(t)|=|\varphi(t')|=1$ for incommensurable $t$ and $t'$ ($t\neq 0$, $t'\neq 0$, $t/t'$ irrational). Show that $P\{X=c\}=1$ for some constant $c$.

26.29 (p. 356) (a) Suppose $X'$ and $X''$ are independent random varibles with values in $[0,2\pi]$, and let $X$ be $X'+X''$ reduced module $2\pi$. Show that the corresponding Fourrier coefficients satisty $c_m=c_m' c_m''$. (b) Show that if one or the other of $X'$ and $X''$ is uniformly distributed, so is $X$.

Answer 1

One interesting idea is the following: Of all distributions on $[0,2\pi]$, the uniform distribution maximizes entropy. So you could try to prove that the averaging operator cannot decrease entropy, then it becomes natural to guess that there exists an fix-point for this iteration of the averaging operator, which should be the maximum entropy distribution. The point of the information that these are not lattice-distributions would be to ensure that we cannot get caught by a fix-point with lower entropy. Google for "entropy central limit theorem" there is even a book with that in the title!

This ideas are related to what physicists call re-normalization theory.

Answer 2

Looks like you would need to figure out the characteristic function of this sum. The problem 26.29 hints at this c.f. converging to that of the uniform distribution, by virtue of the coefficients at non-zero powers of $t$ going to zero. You would need to verify all the regularity conditions, of course.