Learning the exogenous rate but not the distance to criticality in nearly unstable heavy-tailed Hawkes processes

In the nearly unstable heavy-tailed regime of Jaisson and Rosenbaum, where the intensity of a linear Hawkes process converges to a rough square-root Volterra process, we ask what a record of length $T$ reveals about the relative distance to criticality $1-ρ$ and the exogenous rate $μ$. For the first