TITLE: Central Limit Theorem when the sum of the variances is finite
QUESTION [6 upvotes]: Suppose $(x_i)_{i\in\mathbb{N}}$ a set of strictly positive numbers such that $L=\sum_{i\in\mathbb{N}}x_i$ is finite.
Suppose that $(X_i)_{i\in\mathbb{N}}$ is a set independant (real-valued) random variables, each uniformly distributed in $[-x_i;x_i]$.
I am interested in $S_n=\sum_{i=0}^nX_i$ when $n\rightarrow\infty$.
$S_n$ is clearly in $[-L;L]$, so I guess the Central Limit theorem can't apply here ($\sum_{i\in\mathbb{N}}x_i^2$ is finite), so $S_n$ doesn't converge to $\mathcal{N}(0,?)$ since the probability density function of $S_n$ has a finite support ($[-L;L]$).
Can someone help me to find the distribution of $S_n$ when $n\rightarrow\infty$ (is it something like $\mathcal{N}(0,?)$ restricted to $[-L;L]$ ?)
Thanks

REPLY [2 votes]: There is no central limite theorem here. But never mind, this is a wonderful example of $C^\infty$ function nowhere analytics !
https://en.wikipedia.org/wiki/Fabius_function
What are your favorite instructional counterexamples?