Limit laws for UGROW random graphs

Representations are found for a limit law L(Z(k,p))$L\left(Z(k,p)\right)$ obtained from an expanding sequence of random forests containing n$n$ nodes with p∈(0,1]$p\in (0,1]$ a probability controlling bond formation. One implies that Z(k,p)$Z(k,p)$ is stochastically decreasing as k$k$ increases and that norming gives an exponential limit law. Limit theorems are given for the order of component trees. The proofs exploit properties of the gamma function.