A novel representation of the linear birth process with immigration is analysed. The state space of the process is the collection of permutations of the integers, written in a particular cyclic form. The stochastic structure of the model is particularly simple to describe. The results serve to...

Permutation-valued Markov processes provide a convenient way to describe the genealogical structure of certain population models that allow immigration or mutation. Distinct cycles of the permutation correspond to binary branching trees that describe relationships among members of a particular...

Ranked and size-biased permutations are particular functions on the set of probability measures on the simplex. They represent two recently studied schemes for relabelling groups in certain stochastic models, and are of particular interest in describing the limiting behaviour of such models. We...