Behaviour near explosion in continuous-time Galton-Watson trees

Abstract

We study continuous-time Galton--Watson trees whose offspring generating function takes the form \begin{align} \label{eq:gen} f(s) = s - (1-s)^αL(1-s), \end{align} where $α\in (0,1)$ and where $L:[0,1] \to [0,\infty)$ is slowly varying at zero. Processes with this offspring generating function explode in finite time. We observe that, conditional on explosion at time $T$, at each earlier time $t < T$ there is a unique particle $ξ_t$ alive at time $t$ who is an ancestor of all but finitely many particles at the explosion time. We call $(ξ_t)_{t \in [0,T)}$ the spine to explosion, and show that the births off the spine admit a Poissonian description, where they become both more frequent and larger as $t \uparrow T$. We undertake a careful study of the size of the population leading up to explosion, showing in particular that near explosion, the rescaled population is approximately gamma distributed. More generally, conditional on explosion at time $T$, define a stochastic process $Z^\varepsilon := (Z_t^\varepsilon)_{t \in \mathbb{R}}$ by setting \begin{align*} Z_t^\varepsilon := E(\varepsilon e^{-t})N_{T-\varepsilon e^{-t}}, \qquad E(t) = \mathbf{P}(N_t = \infty). \end{align*} We show that as $\varepsilon \downarrow 0$, $Z^\varepsilon$ converges in finite-dimensional distributions to a stationary continuous-state branching process with immigration whose branching mechanism is subcritical and whose immigration mechanism corresponds to spine events. Finally, we show that this limiting process has a Markovian time reversal, and that the coalescent process associated with it is a stochastically time-changed Beta$(2-α,α)$-coalescent.