I have a sum of random variables. 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).

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?