I am interested in the concentration of polynomials of random variables. I have been reading Boucheron, Lugosi, and Massart’s “Concentration inequalities” and they give some references. However, it is a 2013 book, so there naturally isn’t anything that came after that.
Specifically, I am interested in the low-expectation regime: I have a constant-degree polynomial of Bernoulli variables which are all heavily biased towards zero. Moreover, there may not be that many terms and I am trying to set the bias of the Bernoulli variables to be as small as possible while getting sufficient concentration. I am interested also in the lower tail, so things like Janson’s deletion method will not help (or will it?).
I have found “On the concentration of multivariate polynomials with small expectation” from the year 2000. Have there been some developments since then? Something tighter or easier to use? I would also be interested in other recent Kim-Vu type inequalities as they may have reference to what I need in the “related work” section.