Аннотация:
A continuous-time Galton-Watson process is a process initiated by a single particle which lives for a random time $T \sim Exp(\beta)$ for some parameter $\beta>0$ referred to as the branching rate. Upon death the initial particle gives birth to a random number $\xi$ of new particles, where $\xi$ follows some distribution $\mathbb{P}(\xi = k) = p_k$, $k \geq 0$ referred to as the offspring distribution. New particles independently of each other and of the past replicate the initial particle's behaviour. This goes on forever or until there are no particles left in the system. We let $N_t$ be the number of particles in the system alive at time $t$ and $\mathcal{T}_t$ the genealogical tree of the process evolved up to time $t$.
We are interested in the special case of such processes when $\mathbb{E}[\xi] = 1$ called the critical case. It is known that in this case the process eventually becomes extinct, but conditioned to survive to time $t$ it shows interesting behaviour in the limit as $t \to \infty$. For example, conditional on $N_t>0$, the process $\frac{N_t}{t}$ converges in distribution to an Exponential random variable (this goes back to the works of Kolmogorov and Yaglom). Furthermore, conditional on $N_t>0$, the contour process of the tree $\mathcal{T}_t$ converges to a Brownian excursion (this goes back to the works of Aldous) and so the Brownian excursion encompasses the genealogical structure of a critical Galton-Watson process.
In this talk we want to show how Poisson-point-process structure of a Brownian excursion (see, for example, "A guided tour through excursions" by Rogers) can be used to recover the limiting joint distribution of split times of $k$ particles sampled uniformly at random from the population in a critical Galton-Watson process conditioned to survive to time $t$ as $t \to \infty$.
