Suppose $\{\xi_n\}$ are independent random variables such that $|\xi_n|\leq C$, $\forall n$. They also have densities $p_n(x)$, and $|p_n(x)|\leq C$, $\forall n,x$. Is it possible that $\sum\xi_n$ converge in distribution to a uniform distribution?
There is another similar question, on which I am equally clueless:
Suppose $\{\xi_n\}$ are independent random variables uniformly distributed on $[0,1]$, are there real numbers $\{a_n\}$, $\{B_n\}$ such that $0\leq |a_n|\leq 1$, $\forall n$, and $$B_n+\sum_{k=1}^na_k\xi_k\to\zeta$$ weakly, $\zeta$ is uniformly distributed on $[0,2]$?
I feel like proof of weak convergence is usually done with characteristic functions, some form of central limit theorem, or direct computation of distribution functions. But none of these appears to work in these case. Any help is greatly appreciated, thanks.