Persistence and lack of persistence in the PAVD model without higher-order assumptions

Let the PAVD model be the preferential attachment model with vertex death from Definition~. Suppose that the attachment and death-rate sequences $b$ and $d$ satisfy Assumptions~ and~. Let $R$ be the quantity defined in~, assume

$$\limsup_{i\to\infty}d(i)<R,$$

and suppose that $b$ diverges. The persistence conjecture. Persistence should hold in the sense of~ when Assumption~ is not satisfied, whereas lack of persistence should hold in the sense of~ when Assumption~ is satisfied. This conjecture proposes that the higher-order assumptions $\mathcal{K}$ and $\mathcal{K}_\alpha$ are unnecessary for determining persistence or lack of persistence, extending the results beyond the hypotheses used to control the higher-order behaviour of the relevant quantities.