Self-decomposability of α-Cauchy distributions

Abstract

In 2009, Yano, Yano and Yor asked whether the α-Cauchy distributions are self-decomposable or at least infinitely divisible in their study of hitting times for symmetric stable processes. Recently, the first named author proved that α-Cauchy distribution is infinitely divisible if and only if 1 < α≤ 2. In this paper, we prove that α-Cauchy distribution is self-decomposable if and only if α= 2. The proof is based on a new criterion for self-decomposability in the symmetric case. As an application, we show that the first hitting time of a nonzero point by a symmetric stable process of index 1<α<2 starting from zero is not self-decomposable.

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BibTeX RIS

BibTeX

@misc{wang2026selfdecomposability,
  title = {{Self-decomposability of $α$-Cauchy distributions}},
  author = {Min Wang and Sheng Yin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18536},
  eprint = {2609.18536},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18536},
}

RIS

TY  - UNPB
TI  - Self-decomposability of α-Cauchy distributions
AU  - Min Wang
AU  - Sheng Yin
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18536
UR  - https://arxiv.org/abs/2609.18536
AB  - In 2009, Yano, Yano and Yor asked whether the α-Cauchy distributions are self-decomposable or at least infinitely divisible in their study of hitting times for symmetric stable processes. Recently, the first named author proved that α-Cauchy distribution is infinitely divisible if and only if 1 < α≤ 2. In this paper, we prove that α-Cauchy distribution is self-decomposable if and only if α= 2. The proof is based on a new criterion for self-decomposability in the symmetric case. As an application, we show that the first hitting time of a nonzero point by a symmetric stable process of index 1<α<2 starting from zero is not self-decomposable.
ER  -

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