Could someone point me to a way of bounding the tail probability of sums of bernoulli variables each generated by the same distribution but the condition of independence is only partially satisfied. By partially I mean the variables could be divided into subsets of size atmost k such that any variable from two different sets are independent.
Of course one possible technique is that we sum the variables in a set and treat the sums as my new random variables but now the bound of the value we would have on these variables would be k and that's unreasonable in the setting that we have. So could something better be said ?
If needed the corelation between variables belonging to a set could be better characterized.