Universal degree distribution formula for recursive trees with fitnesses

Let $\mathcal{T}$ be a $(\mu,f,\ell)$-RIF tree, and define

$$\alpha=\inf\left\{\lambda>0:\ell\,\mathbb{E}\left[\sum_{n=1}^{\infty}\prod_{i=0}^{n-1}\frac{f(i,W)}{f(i,W)+\lambda}\right]<\infty\right\}.$$

For $\alpha<\infty$, let $N_k(t,B)$ denote the number of degree-$k$ vertices present by time $t$ whose fitness lies in the measurable set $B$. Universal degree distribution conjecture. For every $k\in\mathbb{N}_0$ and measurable set $B$,

$$\frac{N_k(t,B)}{\ell t}\xrightarrow[t\to\infty]{}\mathbb{E}\left[\frac{\alpha}{f(k,W)+\alpha}\prod_{i=0}^{k-1}\frac{f(i,W)}{f(i,W)+\alpha}\mathbf{1}_B\right]$$

almost surely. This conjecture proposes that the degree distribution formula is universal for RIF trees, extending the previously established formula beyond the assumptions used in the paper; the limiting expression also encompasses degenerate degree distributions through the formal limit $\alpha\to\infty$.