On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions

A common approach to simulating a Lévy process is to truncate its shot-noise representation. We focus on subordinators and introduce the remainder process, which represents the jumps that are removed by the truncation. We characterize when these processes are self-similar and show that, in the self-similar case, they can be indexed by a parameter α∈(−∞,1). When α∈(0,1), they correspond to α-stable distributions, and when α=0, they correspond to certain generalizations of the Dickman distribution. Thus, the Dickman distribution plays the role of a 0-stable distribution in this context.