Let $X_i \sim \mathrm{Bernoulli}(p_i)$, let $Y_i = X_i / \mathbf{E} X_i$, and let $S_N = \sum_{i=1}^N Y_i$. ($X_i$ are independent.)
Clearly we have $\mathbf{E}[S_N] = N$. Hoeffding's inequality says that $$ \mathbf{P}\{S_N > N + t\} \leq \exp(-ct^2/P_N), $$ where $P_N = \sum_{i=1}^N 1/p_i^2$ and $c$ is an absolute constant. This isn't great, because if $p_i$ are tiny, then $P_N$ is huge, but on the other hand, when $p_i$ are tiny, we expect very few $X_i$ to be 1, and so the concentration seems as though it shouldn't be as bad as Hoeffding suggests. My question is: can we provide a better concentration inequality for this sum of independent random variables or is this really tight?