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.

This paper presents a study of the intertemporal propagation of distributional properties of phenotypes in general polygenic multisex inheritance models with sex- and time-dependent heritabilities. It further analyzes the implications of these models under heavy-tailedness of traits' initial distributions. Our results suggest the optimality of a flexible asexual/binary mating system. Switching between asexual and binary inheritance mechanisms allows the population effectively to achieve a fast suppression of negative traits and a fast dispersion of positive traits, regardless of the distributional properties of the phenotypes in the initial period.

We find explicit analytical formulae for the time dependence of the probability of the number of Okazaki fragments produced during the process of DNA replication. This extends a result of Cowan on the asymptotic probability distribution of these fragments.

The birth, death and catastrophe process is an extension of the birth–death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

We consider an interacting particle system in which each site of the d-dimensional integer lattice can be in state 0, 1, or 2. Our aim is to model the spread of disease in plant populations, so think of 0 = vacant, 1 = healthy plant, 2 = infected plant. A vacant site becomes occupied by a plant at a rate which increases linearly with the number of plants within range R, up to some saturation level, F1, above which the rate is constant. Similarly, a plant becomes infected at a rate which increases linearly with the number of infected plants within range M, up to some saturation level, F2. An infected plant dies (and the site becomes vacant) at constant rate δ. We discuss coexistence results in one and two dimensions. These results depend on the relative dispersal ranges for plants and disease.

We show that the solution of a quasi-renewal equation with an exponential distribution of the renewals converges at infinity and we compute explicitly the limit, hence generalizing the classical renewal theorem. We apply this result to a stochastic model of DNA replication introduced by Cowan and Chiu (1994).

Kohonen self-organizing interval maps are considered. In this model a linear graph is embedded randomly into the unit interval. At each time a point is chosen randomly according to a fixed distribution. The nearest vertex and some of its nearby neighbors are moved closer to the point. These models have been proposed as models of learning in the audio-cortex. The models possess not only the structure of a Markov chain, but also the added structure of a random dynamical system. This structure is used to show that for a large class of these models, in a strong way, the initial conditions are unimportant and only the dynamics govern the future. A contractive condition is proven in spite of the fact that the maps are not continuous. This, in turn, shows that the Markov chain is uniformly ergodic.

In phylogenetic analysis it is useful to study the distribution of the parsimony length of a tree under the null model, by which the leaves are independently assigned letters according to prescribed probabilities. Except in one special case, this distribution is difficult to describe exactly. Here we analyze this distribution by providing a recursive and readily computable description, establishing large deviation bounds for the parsimony length of a fixed tree on a single site and for the minimum length (maximum parsimony) tree over several sites. We also show that, under very general conditions, the former distribution converges asymptotically to the normal, thereby settling a recent conjecture. Furthermore, we show how the mean and variance of this distribution can be efficiently calculated. The proof of normality requires a number of new and recent results, as the parsimony length is not directly expressible as a sum of independent random variables, and so normality does not follow immediately from a standard central limit theorem.

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.