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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2025-03-01
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| author | Michael Grabchak |
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| description | 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. |
| format | Article |
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| institution | OA Journals |
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| publishDate | 2025-03-01 |
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| spelling | doaj-art-0693a96fea064470a454c6bbf89c3f002025-08-20T01:48:53ZengMDPI AGMathematics2227-73902025-03-0113690710.3390/math13060907On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman DistributionsMichael Grabchak0Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USAA 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.https://www.mdpi.com/2227-7390/13/6/907Lévy processesstable distributionsDickman distributionshot-noise representationself-similar processes |
| spellingShingle | Michael Grabchak On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions Mathematics Lévy processes stable distributions Dickman distribution shot-noise representation self-similar processes |
| title | On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions |
| title_full | On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions |
| title_fullStr | On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions |
| title_full_unstemmed | On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions |
| title_short | On the Self-Similarity of Remainder Processes and the Relationship Between Stable and Dickman Distributions |
| title_sort | on the self similarity of remainder processes and the relationship between stable and dickman distributions |
| topic | Lévy processes stable distributions Dickman distribution shot-noise representation self-similar processes |
| url | https://www.mdpi.com/2227-7390/13/6/907 |
| work_keys_str_mv | AT michaelgrabchak ontheselfsimilarityofremainderprocessesandtherelationshipbetweenstableanddickmandistributions |