  A complete characterization of parabolic self-maps of finite shift is given
in terms of their Herglotz's representation. This improves a previous result
due to Contreras, D\'iaz-Madrigal, and Pommerenke. We also derive some
consequences for the rate of convergence of these functions to their
Denjoy-Wolff point, improving a related result of Kourou, Theodosiadis, and
Zarvalis for the continuous setting.
