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

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-...

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Bibliographic Details
Main Author: Michael Grabchak
Format: Article
Language:English
Published: MDPI AG 2025-03-01
Series:Mathematics
Subjects:
Lévy processes
stable distributions
Dickman distribution
shot-noise representation
self-similar processes
Online Access:https://www.mdpi.com/2227-7390/13/6/907
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Summary: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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, they correspond to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable distributions, and when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, they correspond to certain generalizations of the Dickman distribution. Thus, the Dickman distribution plays the role of a 0-stable distribution in this context.
ISSN:2227-7390

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