I have a post Tail bound on difference of shifted binomials answered before and now I want to consider I slightly generalization of it which can't be solved using the methods in the previous thread.
I want to derive an upper bound on $\mathsf{Pr}\{\sum_{i=1}^n(\frac{X_i}{L}-\frac{1}{2})^2 \leq \sum_{i=1}^n(\frac{Y_i}{L}-\frac{1}{2})^2\}$ where $X_i,Y_i$ are all independent and $X_i\sim$ Bin($L,p$) where $p\neq \frac{1}{2}$, and $Y_i\sim$ Bin($L,\frac{1}{2}$).
Intuitively, the probability should be vanishing with both $n$ and $L$.
If I use my original method stated in the pervious post (which is Chernoff bound combined with Hoeffding's Lemma), I can only get the upper bound $\exp\{-16n(\frac{L-1}{L})^2(p^2+\frac{1}{4})^2\}$ which is vanishing with $n$ but not with $L$.
If I use the method by Gabriel Romon in the pervious post, the upper bound would become $4n\exp\{-\frac{L}{2}(p-\frac{1}{2})^2\}$ which is vanishing with $L$ but not with $n$. (Since we use the union bound.)
Thanks in advanced.