The number of descendants in a random directed acyclic graph

We consider a well-known model of random directed acyclic graphs of order n$$ n $$, obtained by recursively adding vertices, where each new vertex has a fixed outdegree d > 2$$ d\geqslant 2 $$ and the endpoints of the d$$ d $$ edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number X(n)$$ {X}<^>{(n)} $$of vertices that are descendants of n$$ n $$. We show that X(n)/n(d-1)/d$$ {X}<^>{(n)}/{n}<^>{\left(d-1\right)/d} $$ converges in distribution; the limit distribution is, up to a constant factor, given by the d$$ d $$th root of a Gamma distributed variable with distribution Gamma(d/(d-1))$$ \Gamma \left(d/\left(d-1\right)\right) $$. When d=2$$ d=2 $$, the limit distribution can also be described as a chi distribution chi(4)$$ \chi (4) $$. We also show convergence of moments, and find thus the asymptotics of the mean and higher moments.