Note: I divide this question to two separated question not to be duplicate version.
I am looking for tail bounds (preferably exponential) for the sum of dependent and bounded random variables.
Consider $K_{ij}=\sum_{r=1}^N\sum_{c=1}^N W_{ir}W_{jc}$ where $i \neq j$, $W\in \{+1, -1\}$ and $W$ are i.i.d. random variables $\operatorname{Bernoulli}(0.5)$. How can I obtain an exponential bound over $Pr[K_{ij} \geq \epsilon] \leq \exp(?)$ where $\epsilon$ is a positive value.
Answer: $K_{ij}$ can be considered as the multiplication of two independent Binomial random variables, i.e., $K_{ij}=\left(\sum_{r=1}^N W_{ir}\right)\left(\sum_{c=1}^N W_{jc}\right)$, then the moment generating function can be evaluated and then by using the Chernoff we can have a tail bound for $K_{ij}$.
Thanks a lot in advance.