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