Consider the random variable $\delta_1$ with the following probability distribution, where we have:
$\mathbb{P}(\delta_1=0) = 1 - 2 \delta$, $\mathbb{P}(\delta_1=1)= \delta$, $\mathbb{P}(\delta_1=-1)= \delta$, where $\delta \ll 1$ is a small number.
Let $(\delta_i)_{1 \leq i \leq n}$ by an i.i.d. collection of random variables distributed like $\delta_1$. I am looking for a good concentration bound for expression $\mathbb{P}(\sum_{i=1}^{n} a_i \delta_i \geq t)$.
A try with Hoeffding's inequality gives that $\mathbb{P}(\sum_{i=1}^{n} a_i \delta_i \geq t) \leq \exp(-t^2/\sum_{i \geq 1} a_i^2)$. This bound, while exponential does not reflect that the random variables $\delta_i$ are highly concentrated around their mean so we should expect a bit more concentration of measure (Rademacher random variables also give the same bound although they have much higher variance).
I tried using a chernoff bound type strategy which seems to yield something like $\mathbb{P}(\sum_{i=1}^{n} a_i \delta_i \geq t) \leq \exp(-(t^2/\sum_{i \geq 1} a_i^2) +n \log \delta)$.
My question is: Is this the best possible bound I can get? I wondering if I can do any better than this. If it makes any difference, in this particular case we have $\delta = n^{- \epsilon}$ for some fixed $\epsilon>0$.
Edit: I would very much like to have something like $\mathbb{P}(\sum_{i=1}^{n} a_i \delta_i \geq t) \leq \exp(-(t^2/\delta^{O(1)} \sum_{i \geq 1} a_i^2)$, but I am skeptical as to whether this would be possible. In general, is there a way to tell if a concentration bound is 'sharp'? How would one go about proving this?