For random variables with values on binary metric trees, the definition of the expected value can be generalized to the notion of a barycenter. To estimate the barycenter from tree-valued data, the so-called inductive mean is constructed recursively using the weighted interpolation between the current mean and a new data point. We show the strong consistency of the inductive mean, but also that it, somewhat peculiarly, converges towards the true barycenter with different rates, and asymptotic distributions depending on the small variations of the underlying distribution.

In bioinformatics, the notion of an ‘island’ enhances the efficient simulation of gapped local alignment statistics. This paper generalizes several results relevant to gapless local alignment statistics from one to higher dimensions, with a particular eye to applications in gapped alignment statistics. For example, reversal of paths (rather than of discrete time) generalizes a distributional equality, from queueing theory, between the Lindley (local sum) and maximum processes. Systematic investigation of an ‘ownership’ relationship among vertices in ℤ2 formalizes the notion of an island as a set of vertices having a common owner. Predictably, islands possess some stochastic ordering and spatial averaging properties. Moreover, however, the average number of vertices in a subcritical stationary island is 1, generalizing a theorem of Kac about stationary point processes. The generalization leads to alternative ways of simulating some island statistics.

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. H is non-lattice with mean value λ –1, and if the traffic intensity ρ = λμ –1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn, where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and πn for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of πn to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r0 > 1 with , then the exact convergence rate of πn to π is characterized by r0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between πn and π based on the moments of A.

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ n at the arrival instant. Assuming Harris ergodicity of {θ n} and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.