Open Access

It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman's two-parameter Chinese restaurant process, tree-growth models associated with Mallows' ϕ model of random permutations and with Schützenberger's non-commutative q-binomial theorem, and a construction due to Luczak and Winkler that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail σ-fields.

A perpetuity is a real valued random variable which is characterised by a distributional fixed-point equation of the form X=AX+b, where (A,b) is a vector of random variables independent of X, whereas dependencies between A and b are allowed. Conditions for existence and uniqueness of solutions of such fixed-point equations are known, as is the tail behaviour for most cases. In this work, we look at the central area and develop an algorithm to approximate the distribution function and possibly density of a large class of such perpetuities. For one specific example from the probabilistic analysis of algorithms, the algorithm is implemented and explicit error bounds for this approximation are given. At last, we look at some examples, where the densities or at least some properties are known to compare the theoretical error bounds to the actual error of the approximation. The algorithm used here is based on a method which was developed for another class of fixed-point equations. While adapting to this case, a considerable improvement was found, which can be translated to the original method.