Limit laws for UGROW random graphs

Representations are found for a limit law L (. Z (. k, p) ) obtained from an expanding sequence of random forests containing n nodes with p ∈ (0, 1] a probability controlling bond formation. One implies that Z (. k, p) is stochastically decreasing as 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. © 2013 Elsevier B.V.