My congratulations to Mr. Leigh J. Halliwell on this paper that clearly presents the mathematics of excess losses with an interesting example. I agree with him that the mathematics of excess losses is beautiful and powerful. However, the mathematics of excess losses also contains several subtle points that are not mentioned in the paper. This discussion note complements the article by clarifying some of these points. To be clear, it is not my intention to be critical of Mr. Halliwell. The purpose of this note is two-fold:
1. To clarify some important hidden points in the mathematics of excess losses;
2. To give references to some uncredited results.
For those ambitious actuaries who want to dig deeper for a full understanding of the rigorous mathematics of excess losses, this note also provides some directions for further studies.
For the convenience of readers, we will adopt the notations in Halliwell (2013). Throughout this note, X will denote a nonnegative random variable. F and G will denote the cumulative distribution function (CDF) and survival function of X, respectively.
