I have a sum of random variables, they are not identically distributed, but even that case would be interesting. They are not necessarily independent, but I already have a concentration bound for the individual random variables (that I got using one of the standard methods, such as Chernoff bound, Method of bounded differences, Kim-Vu inequality or similar). I care about the relative size of the deviations (that is, as a fraction of the expectation).

Is there anything that I can say about the concentration of the sum in general? Is there a method how to solve this?

I could try to simply explicitly calculate the deviation probabilities by convolution/characteristic functions but that seems rather complicated. Is there an easier (and presumably less precise) way?

I could also estimate the probability that each random variable is deviated by a $\epsilon$-fraction and then use union bound. However, this seems rather wasteful and I cannot think of an example where this would be tight (or even nearly tight)