We consider a version of a classical concentration inequality for sums of independent, isotropic random vectors with a mild restriction on the distribution of the radial part of these vectors. The proof uses a little Fourier analysis, the Laplace asymptotic method and a conditioning idea that traces its roots to some of the original works on concentration inequalities.
"On a concentration inequality for sums of independent isotropic vectors." Electron. Commun. Probab. 17 1 - 8, 2012. https://doi.org/10.1214/ECP.v17-2063