On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions - Michael Grabchak (Mathematics)
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 <semantics>α∈(−∞,1)</semantics>. When <semantics>α∈(0,1)</semantics>, they correspond to <semantics>α</semantics>-stable distributions, and when <semantics>α=0</semantics>, they correspond to certain generalizations of the Dickman distribution. Thus, the Dickman distribution plays the role of a 0-stable distribution in this context.