The Galton-Watson tree conditioned on its height (Q2769678)

Let \(Z_n\) be a Galton-Watson branching process, with non-trivial offspring distribution \(p_k\). Let \(H\) be the height of the associated tree, i.e. \(H=\inf\{n:Z_{n+1}=0\}\). The main theorem of the authors states that, for arbitrary \(p_k\)'s, the distribution of \(Z_n\) conditioned by \(H=n\), converges in a strong sense to the law \(\nu(k)=p_0^k\pi_k\), \(k\geq 1,\) where \(\pi_k=\lim_{n\rightarrow\infty}P(Z_n=k)/P(H=n)\) is also characterized as a solution of an infinite linear system. A weaker result is given for the distribution of \(Z_H\) conditioned by the total population size \(N=\sum_{k=0}^\infty Z_k\); here in the critical, finite-variance case, one gets NEWLINE\[NEWLINE \lim_{n\rightarrow\infty}P(Z_H=k\mid N\geq n)=\nu(k),\quad k\geq 1.NEWLINE\]NEWLINE The proof of these results is based on the fact that a G-W tree conditioned by its height, can be decomposed along the line of descent of the left-most particle of generation \(H\), in a collection of independent trees (conditioned by different events), yielding a representation of the final generation size as a sum of independent increments.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].