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

Author

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  • Michael Grabchak

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

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.

Suggested Citation

  • Michael Grabchak, 2025. "On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions," Mathematics, MDPI, vol. 13(6), pages 1-15, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:6:p:907-:d:1608038
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    Cited by:

    1. Michael Grabchak & Xingnan Zhang, 2026. "On the simulation of general multivariate gamma distributions using Dickman approximations," Computational Statistics, Springer, vol. 41(3), pages 1-25, April.

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