Tilting parameters for non-eigenvector-aligned type conditioning

Determine whether, for a multitype Bienaymé offspring distribution and prescribed type-count asymptotics x_i(n)=\bar{a}_i n+O(\sqrt{n}), one can choose exponential tilting parameters \theta_1,\ldots,\theta_{K+K'} so that the tilted offspring distribution satisfies the assumptions of the type-conditioning scaling theorem and has a nonnegative left 1-eigenvector whose coordinates equal \bar{a}_i for every conditioned type i.

Background

The paper proves a Brownian Continuum Random Tree scaling limit when the prescribed numbers of vertices of the conditioned types satisfy x_i(n)=a_i n+O(\sqrt{n}), where (a_i) is the relevant normalized left Perron–Frobenius eigenvector of the offspring mean matrix. The authors then discuss exponential tilting of the offspring distribution, which preserves the conditional distribution under fixed type counts while changing the mean matrix and its eigenvector.

The unresolved issue is whether this tilting procedure can generally align the left 1-eigenvector with an arbitrary target direction (\bar{a}_i) describing the prescribed type proportions, while retaining the assumptions needed for the scaling theorem. The paper establishes such a result only under the additional hypotheses of irreducibility, entire generating functions, nonlocalization, and strong accessibility.

References

Hence, the question is, if~eq:cccond is not satisfied and instead we have

x_i(n) = \bar{a}_i n + O(\sqrt{n})

for constants $\bar{a}i \ge 0$, $i \in I$, can we find parameters $\theta_1, \ldots, \theta{K+K'}$ such that ${\bm{\theta}$ satisfies the assumptions in Theorem~\ref{thm:bytype} and such that the left $1$-eigenvector $(a_i{\bm{\theta}){1 \le i \le K+K'}$ of $A{\bm{\theta}$ with non-negative coordinates normalized to $\sum_{i=1}{K} a_i{\bm{\theta} = 1$ (whose existence and uniqueness is then guaranteed) satisfies $a_i{\bm{\theta} = \bar{a}_i$ for all $i \in I$?

— Scaling limits of multitype Bienaymé trees  (2507.23241 - Addario-Berry et al., 31 Jul 2025) in Section 1, subsection “Main result when conditioning by types”