Infinite divisibility of α-Cauchy distributions

2025/12/29 by Min Jeong Wang, Wang, Min
Decision Sciences · Mathematics · #33E12 #60E05 #60E07 #60E10 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications

paper · doi:10.48550/arxiv.2512.23164

openalex publication_date 2025/12/29 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28

Abstract

In 2009, Yano, Yano and Yor proposed the question of studying the infinite divisibility of the α-Cauchy variable C_α for α> 1. The particular case C₂ is the well-known standard Cauchy variable, which is infinitely divisible and indeed stable. For α≠ 2, the infinite divisibility of C_α is previously unknown. In this paper, we prove that C_α is infinitely divisible if and only if 1 < α≤ 2.

Citations

• On the log-concavity of the Wright function
• Selfdecomposability of Weak Variance Generalised Gamma Convolutions
• On the law of homogeneous stable functionals
• On the infinite divisibility of inverse Beta distributions
• HCM Property and the Half-Cauchy Distribution
• On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes

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