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Infinite divisibility of powers of positive stable random variables

  • Min Wang
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Abstract

Let Zα be a positive α-stable random variable, where 0<α<1. We prove that Zαp is infinitely divisible for every p>0. Combining this result with Jedidi and Simon's characterization of infinite divisibility of negative powers of Zα, we obtain, for p≠0, Zαp is infinitely divisible Longleftrightarrow p∈(−∞,−α/(1−α)]∪(0,∞). The proof relies on Kanter's factorization and a general product property: if X is a mixture of exponential distributions, W is a generalized gamma convolution, and X and W are independent, then (X+c)W is infinitely divisible for every c≥ 0.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026infinite,
  title = {{Infinite divisibility of powers of positive stable random variables}},
  author = {Min Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31802},
  eprint = {2609.31802},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31802},
}
RIS
TY  - UNPB
TI  - Infinite divisibility of powers of positive stable random variables
AU  - Min Wang
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31802
UR  - https://arxiv.org/abs/2609.31802
AB  - Let Z_α be a positive α-stable random variable, where 0<α<1. We prove that Z_α^p is infinitely divisible for every p>0. Combining this result with Jedidi and Simon's characterization of infinite divisibility of negative powers of Z_α, we obtain, for p≠0, Z_α^p is infinitely divisible Longleftrightarrow p∈(−∞,−α/(1−α)]∪(0,∞). The proof relies on Kanter's factorization and a general product property: if X is a mixture of exponential distributions, W is a generalized gamma convolution, and X and W are independent, then (X+c)W is infinitely divisible for every c≥ 0.
ER  -