Realizability of all stable offspring distributions by subordination

Determine whether every critical offspring distribution in the domain of attraction of an ((α+1)/2)-stable law can be realized as the offspring distribution of a subordinate tree generated from a critical offspring distribution in the domain of attraction of an α-stable law, with a suitable centered jump distribution satisfying the paper’s assumptions.

Background

The paper proves that if the original critical Bienaymé offspring distribution μ belongs to dom(α), and the centered jump distribution satisfies the stated tail and bounded-positive-jump assumptions, then the subordinate tree is a Bienaymé tree whose offspring distribution ν belongs to dom((α+1)/2). This establishes a one-way transformation of stable domains under subordination.

The authors note that subordinate offspring distributions necessarily have infinite variance, so not every Bienaymé offspring distribution can arise in this way. The unresolved converse question is whether every critical distribution ν in the relevant ((α+1)/2)-stable domain can be obtained from some suitable μ and jump distribution.

References

We do not know if every critical offspring distribution $\nu \in \mathrm{dom}(\frac{\alpha+1}{2})$ may be realized by choosing a well-suited $\mu \in \mathrm{dom}(\alpha)$.

— Subordination of discrete snakes  (2610.02115 - Aurillard et al., 1 Oct 2026) in Section 2, subsection “Subordination of (sub-)critical snakes,” immediately after Theorem 2.5 (Theorem~\ref{thm:NewOffspringStable})

Finally, for $\kappa<\frac12$, the equation appearing to invert $G_{\mu}{-1}$ cannot be algebraically solved hence we were not able to prove the existence of a corresponding $\mu$.

— Subordination of discrete snakes  (2610.02115 - Aurillard et al., 1 Oct 2026) in Section 5, subsection “An explicit example (Example~\ref{ex:ExampleAndrea}),” final remark