Variations and extensions of the Gaussian concentration inequality, Part II

We prove concentration inequalities for $f\left( X\right) $ about its median, where $X$ is a random vector in $\mathbb{R}^n$ with independent heavy tailed coordinates of Weibull or power type, and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a locally Lipschitz function. This paper is part of a series of four papers, Part I, Part II and two supporting papers. It can be read independently of Part I.
Comment: 14 pages. The original paper, which was uploaded as a separate submission (arXiv:1812.10938), has been split into two papers, Part I and Part II; this is Part II