On existence and uniqueness of univariate Stein kernels

We completely characterize the class of univariate distributions allowing for a Stein kernel and illustrate our result by means of some concrete distributions. Moreover, we apply our findings to prove a quantitative version of the central limit theorem with optimal rate $n^{-1\mathrm{/}2}$ in total variation distance for i.i.d. random variables whose distribution belongs to that class.