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

Michael Grabchak ()
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Michael Grabchak: Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA

Mathematics, 2025, vol. 13, issue 6, 1-14

Abstract: 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.

Keywords: Lévy processes; stable distributions; Dickman distribution; shot-noise representation; self-similar processes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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