Let C_alpha denote the alpha-Cauchy distribution for 1<alpha≤ 2. Determine exactly for which values of alpha the law C_alpha is self-decomposable; that is, for every c∈(0,1), whether there exists a probability law nu_c such that C_alpha=D_cC_alphanu_c, where D_c denotes scaling by c. The claimed classification is C_alpha is self-decomposable if and only if alpha=2. The associated application asks whether, for a symmetric stable process of index 1<alpha<2 started at 0 and a nonzero point x, the first hitting time T_x=inft≥ 0:X_t=x is self-decomposable; the cited preprint claims that it is not.

Self-decomposability of α-Cauchy distributions

Let $\mathcal{C}_{\alpha}$ denote the $\alpha$ -Cauchy distribution for $1<\alpha\leq 2$ . Determine exactly for which values of $\alpha$ the law $\mathcal{C}_{\alpha}$ is self-decomposable; that is, for every $c\in(0,1)$ , whether there exists a probability law $\nu_c$ such that $\mathcal{C}_{\alpha}=D_c\mathcal{C}_{\alpha}*\nu_c$ , where $D_c$ denotes scaling by $c$ . The claimed classification is $\mathcal{C}_{\alpha}$ is self-decomposable if and only if $\alpha=2$ . The associated application asks whether, for a symmetric stable process of index $1<\alpha<2$ started at $0$ and a nonzero point $x$ , the first hitting time $T_x=\inf\{t\geq 0:X_t=x\}$ is self-decomposable; the cited preprint claims that it is not.

References

Primary source

arXiv

Additional references

Self-decomposability of α-Cauchy distributions — arXiv — Min Wang, Sheng Yin

probability stochastic-processes

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A new unrefereed preprint claims to settle the full classification, but its result has not yet been independently verified.

The problem asks for the complete self-decomposability classification of $\alpha$ -Cauchy distributions, including associated hitting-time applications. Yano, Yano, and Yor posed the underlying questions in 2009.

Known results

Infinite divisibility was characterized in December 2025: $\mathcal{C}_{\alpha}$ is infinitely divisible exactly when $1<\alpha\le 2$ ; self-decomposability was explicitly left open.

September 2026 claimed classification

Min Wang and Sheng Yin’s preprint claims a full self-decomposability classification and applications to hitting times of symmetric stable processes. The claim is currently unrefereed and has not been independently verified.

Current status (as of September 2026): a complete solution is claimed in an unrefereed preprint, while independent verification and refereeing remain outstanding.

Sources

No solutions have been posted yet.