In this paper we improve the rate function in the McDiarmid concentration inequality for separately Lipschitz functions of independent random variables. In particular the rate function tends to infinity at the boundary. We also prove that in some cases the usual normalization factor is not adequate and may be improved.
"On McDiarmid's concentration inequality." Electron. Commun. Probab. 18 1 - 11, 2013. https://doi.org/10.1214/ECP.v18-2659