A concentration problem Consider $N$ variables $X_1,X_2,\ldots,X_N$ with $\Pr(X_i=a_i)=\Pr(X_i=-a_i)=1/2$. Here $a_1,a_2,\ldots,a_N\in [0,1]$.
Does there exist some concentration result about $\sum_{i=1}^N X_i$?

If $a_i$ are deterministic, than you can think about new random variables  $Y_i=\frac{X_i}{a_i}$ and you already have the lower bound for this.
