In a previous paper [ibid. 11, 158-167 (1983; Zbl 0503.60035), the author introduced a concept of ''dominating'' points to establish a nice representation formula and improved asymptotics for the probabilities of large deviations of random walks on \({\mathbb{R}}^ d\). In this short addendum it is shown that the proof can be simplified making use of convexity instead of fixed point and probabilistic arguments.